CLEP Pre-Calculus. Section 1: Time 30 Minutes 50 Questions. 1. According to the tables for f(x) and g(x) below, what is the value of [f + g]( 1)?

Size: px
Start display at page:

Download "CLEP Pre-Calculus. Section 1: Time 30 Minutes 50 Questions. 1. According to the tables for f(x) and g(x) below, what is the value of [f + g]( 1)?"

Transcription

1 CLEP Pre-Calculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (non-cas) is allowed to be used for this section.. According to the tables for f(x) and g(x) below, what is the value of [f + g]( )? (A) (B) 4 (C) 7 (D) 0 (E) x f(x) x g(x) According to the graphs of f(x) and g(x) shown, what is the value of [g f]()? (A) (B) 0 (C) (D) (E) Cannot be determined Copyright 006 Peterson's CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

2 CLEP Pre-Calculus. Alex's grades are shown in the table below for pre-calculus tests. If each test is weighted equally, what is the lowest grade that Alex can score on the fifth and final test to have an average of at least 90%? (A) 89.5% (B) 90% (C) 90.5% (D) 9% (E) 94% Test Grade # 85% # 9% # 86% #4 94% 4. Which of the following could be an equation for the hyperbola shown? (A) (B) (C) (D) (E) ( y 5) ( x + 4) a b ( x + 4) ( y 5) a b ( y + 5) ( x 4) a b y x 5 6 ( y + 5) ( x + 4) + a b 5. Let f(x) x 6, and let g(x) 5x + 4. Which of the following is equivalent to [f g](x)? (A) 0x 8x 4 (B) 0x + (C) x (D) 0x 4 (E) 0x x - 4 Copyright 006 Peterson's CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

3 CLEP Pre-Calculus 6. Find S so that the sum of the areas of the three figures shown is less than 4. (A) < S < 6 (B) S < or S < 6 (C) 6 < S < (D) S < (E) Cannot be determined 7. Suppose cos( θ ). What is the value of tan( θ )? 8 (A) 7 (B) 65 (C) 7 (D) 7 (E) 8 8. Find the exact value for x if 4 x 6 4. (A) (B) (C) (D) (E) log(0) log(4) log(0) log(4) log(0) log(4) log(4) log(0) log(4) log(0) Copyright 006 Peterson's CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

4 CLEP Pre-Calculus 9. What is the circumference of the circle with equation (x ) + (y + ) 49? (A) (B) (C) (D) (E) 7π 4π 8π 49π 60π 0. An ellipse has an equation of ellipse? ( x ) ( y + ) +. Which statement is true about this 4 9 (A) (B) (C) (D) (E) The center is at (, ) with a horizontal major axis. The center is at (, ) with a vertical major axis. The center is at (-, ) with a horizontal major axis. The center is at (-, ) with a vertical major axis. The center is at (-, -) with a vertical major axis.. Find the range of the function h( x) + 5. x (A) (,) (B) (5, ) (C) (, ) (D) (,5) (E) (, ). Find three functions, f(x), g(x), and h(x), such that [ f g h]( x) F( x ) if F( x). cos( x) + 5 (A) (B) (C) (D) (E) f ( x), g( x) x + 5, h( x) cos( x) x f ( x) cos( x), g( x) x, h( x) x f ( x) cos( x), g( x) x +, h( x) x 5 f ( x), g( x) x 5, h( x) sin( x) x f ( x), g( x) x 5, h( x) cos( x) x Copyright 006 Peterson's 4 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

5 CLEP Pre-Calculus. If 5x + 4y 6 and 6x y 9, what is the value of x +? 4. For each of the functions, indicate if the function is even, odd, or neither. Function Even Odd Neither f(x) 4x g(x) 5sin(θ) h(x) x x If x 6, what is the value of x? (A) 4 4 (B) 8 (C) 6 (D) 4 (E) Which of the following equations has a y-intercept of? (A) y x (B) y x + (C) y x + x (D) y x + 7 (E) y sin(x) 7. Let f(x) x, and let the values of g(x) be as shown in the table: What is the value of f(g())? (A) (B) (C) 4 (D) (E) 5 x g( x ) Copyright 006 Peterson's 5 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

6 CLEP Pre-Calculus 8. If cot( θ ) p, 6 what is the value of csc(θ)? (A). (B) (C) (D) (E) 6 + p. 6 ( p + 6)( p 6) p p 6 6 p 9. Find y if 5y tan(0 ) 0. (A) (B) (C) (D) 4 (E) 6 0. The graphs of f(x) and g(x) are shown here. What is the value of g(f(0))? (A) 0 (B) (C) (D) (E) Cannot be determined Copyright 006 Peterson's 6 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

7 CLEP Pre-Calculus. Which of the following could be an equation for the parabola with its vertex at the point ( 6, 8) and a vertical axis? (A) (x + 6) c(y 8) (B) (x 6) c(y + 8) (C) (y + 6) c(x 8) (D) (y 6) x(c + 8) (E) (y 6) x(c 8). Let g(x) x + x 6, and let h(x) x + 5x 8. What is g(h(x))? (A) x 4 + x x 57x + 4 (B) 4x 4 + 0x x 65x + 4 (C) x + 8x 4 (D) x 4 + x 5x 54x + 48 (E) 4x 4 + 0x x 65x + 4. If cos( θ ) z, what is the value of sin( θ )? (A) (B) (C) (D) (E) z 4 + z + z 4 z ( z)( + z) Copyright 006 Peterson's 7 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

8 CLEP Pre-Calculus 4. Which of the following is true about the inverse of the function shown in the graph below? (A) (B) (C) (D) (E) The domain of the function is all real numbers. The range of the function is all positive real numbers. The inverse is not a function. The inverse is one-to-one. Both B and D 5. What is the value of sin(50 )? (A) (B) (C) (D) (E) + Copyright 006 Peterson's 8 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

9 CLEP Pre-Calculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. No calculator is allowed for this section. 6. Which of the following is NOT true about the function represented in the table? x f(x) (A) The function is increasing as x moves from 0 to 5. (B) The graph intersects the x-axis at 6. (C) The slope as x moves from 0 to 5 is. (D) Both B and C. (E) All of the above 7. Let h(x) be the function represented in the table, and let g(x) be the function shown in the graph. What is the value of g(h())? x h(x) (A) 0 (B) (C) 4 (D) 5 (E) 7 Copyright 006 Peterson's 9 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

10 CLEP Pre-Calculus 8. Jeffrey s job pays him $0 per hour and 5% commission on all of his sales. Last week, Jeffrey worked 40 hours and made x dollars in sales. Which of the following expresses the total amount of money that Jeffrey made last week as a function of x? (A) M x (B) M 40x + 00 (C) M 0x (D) M x (E) M 0 +.5x 9. Two buildings facing each other are separated by a distance of 40 feet. From the top of the first building, the angle of depression of the second building s base is 60, and the angle of depression of the top of the second building is 45. What is the height of the second building? (A) 40( ) (B) 40 (C) 40 (D) 40( ) (E) Not enough information to solve this problem 0. Let f(x) x 9x + 7, and let g(x) x 4x 0. What is the value of f(g())?. Michelle is flying a kite that is 8 feet high. If the string of the kite forms a 60 angle with the ground, and Michelle is holding the kite feet off the ground, what is the length of the string? Write your answer to the nearest hundredth of a foot. Copyright 006 Peterson's 0 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

11 CLEP Pre-Calculus. According to the graphs of f(x) and g(x) below, what is the domain of [f + g](x)? (A) (, ) (B) [, ) (C) (, ) (D) (,) (E) [, ). The length of a rectangle is 4 more than times the width. Which of the following is an equation for the length of the diagonal of the rectangle in terms of the width? (A) (B) (C) d 4 + w d 4w + 6w + 6 d 5w + 6w + 6 (D) d 5w + 6w + 6 (E) d 4w + 6w Let f(x) x 4x 5, and let What is the value of [f g](5)? (A) 5.68 (B) (C) 49.8 (D) 50 (E) 50. Copyright 006 Peterson's CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

12 CLEP Pre-Calculus 5. What is the range of the function y cos(x)? (A) [, ] (B) all real numbers (C) all rational numbers (D) [0, ] (E) [-, 0] 6. Suppose sin( θ)cos( θ ). What is the approximate value of θ? 4 (A) (B) 4 (C) 4 (D) 49 (E) 5 7. Suppose the volume of a cylinder is 0 cubic inches. Which of the following is an expression of the surface area of the cylinder in terms of its radius? (A) (B) (C) 60 S r 60 S π ( r) + r 0 S π( ) r (D) 0 S π ( r) + π( r ) (E) S π(r) 8. The sum of all three sides of the right triangle shown is. Find the lengths of its two legs. Copyright 006 Peterson's CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

13 CLEP Pre-Calculus f 9. According to the tables of values for f(x) and g(x), what is the value of ()? g (A) (B) (C) (D) 6 (E) 7 x f(x) g(x) Find all values of q that satisfy the system of equations shown: (A) -.44,.56 (B).8, 4.07 (C) 0.56, 4.4 (D).4, 0.8 (E).44,.4 p + pq q 4 4 p + q 6 4. Let f(x) x + 6x 7, and let g(x) x 4x +. What is the value of f ()? g (A) (B) (C) (D) (E) Copyright 006 Peterson's CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

14 CLEP Pre-Calculus 4. Find a range of values for p if 4 5p 4. (A) p 4 (B) 8 4 p 5 (C) p 4 or (D) 4 p (E) p p According to the table of values for f(x) and g(x), what is the value of [ f g ]()? (A) (B) 0 (C) (D) 4 (E) 6 x f(x) g(x) A right triangle has legs of length 6 inches and 8 inches. What is the measure of the angle opposite the 6-inch leg? (A) 48.6 (B) 4.4 (C) 5. (D) 6.87 (E) Not enough information to solve Copyright 006 Peterson's 4 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

15 CLEP Pre-Calculus 45. Which of the following values for x and y satisfies the system of equations shown? (A) 0 4 x, y (B) 94 0 x, y (C) 9 x, y 4 (D) 8 4 x, y (E) x -, y x + 4y 0 x 5y A woman is walking along a straight road. She notices the top of a building subtending an angle of 0 with the ground at the point where she is standing. If the building is 50 feet tall, how far is the woman from the building? (A) (B) (C) (D) (E) 86.6 feet 8.9 feet 57.7 feet 00 feet 9. feet 47. What is the value of sin(75 )? (A) (B) (C) (D) (E) Copyright 006 Peterson's 5 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

16 CLEP Pre-Calculus 48. A number x is first decreased by 5%, and then the result is increased by 0%. Which of the following functions could be used to determine the final result? (A) f(x) 0.5x (B) f(x) 0.85x (C) f(x) 0.75x (D) f(x) 0.5x (E) f(x) 0.04x Copyright 006 Peterson's 6 CLEP is a registered trademark of the College Entrance Examination Board, which was not involved in the production of and does not endorse this product.

17 CLEP Pre-Calculus. The correct answer is C. The algebraic combination, [f + g](x), is defined as f(x) + g(x). According to the table, f( ) and g( ) 4. Now find the sum: [ f + g]( ) f ( ) + g( ) + 4. The correct answer is A. The algebraic combination [g f](x) is defined as g(x) f(x). According to the graphs, g() 0, and f(). Now subtract to find the difference: 7 [ g f ]() g() f () 0. The correct answer is D. The average is calculated by dividing the sum of the test scores by the number of tests. Let x equal the score on the fifth and final test: sum of scores Average number of tests x x 5 Now plug in 90 for the average and use an inequality sign to represent the situation in terms of x: 58 + x x 90(5) (5) x x 58 9 x 4. The correct answer is A. The standard form of the equation for a hyperbola centered at the ( y k) ( x h) point (h, k) and with a vertical axis is. The hyperbola shown is centered at a b ( y 5) ( x + 4) the point ( 4, 5), so an equation for the hyperbola is. a b ANSWER KEY - Page 7

18 Recovered Master File (08/08/04 : AM) 5. The correct answer is E. The algebraic combination [f g](x) is defined as f(x) g(x). Multiply the equation for f(x) by the equation for g(x) and combine like terms to find the simplified product: [ f g]( x) f ( x) g( x) (x 6) (5x + 4) 0x + 8x 0x 4 0x x 4 6. The correct answer is C. Write an equation for the sum of the area of the three figures. Then solve the inequality for S: Area (S)(S) + ()(S) ()(4) < 4 S S + 4S + < 4 + 4S < 0 ( S )( S + 6) < 0 In order for (S )(S + 6) to be less than zero, one of the factors must be negative, and one must be positive: ( S ) > 0 and ( S + 6) < 0 S > and S < 6 OR ( S ) < 0 and ( S + 6) > 0 S < and S > 6 Since the first solution is impossible, the value of S must lie between 6 and. 7. The correct answer is C. Since sec( θ ), use the identity tan (θ) + sec (θ): cos( θ) ANSWER KEY - Page 8

19 Recovered Master File (08/08/04 : AM) sec( θ ) cos( θ) tan ( θ ) + sec ( θ) tan ( θ ) + (8) tan ( θ ) tan ( θ ) 6 tan( θ ) The correct answer is C. This equation can be solved using the logarithmic function: x x 4 0 x log(4 ) log(0) xlog(4) log(0) x log(0) log(4) 9. The correct answer is B. The standard form for the equation of a circle centered at (h, k) and with radius r is (x h) + (y k) r. In this case, the radius of the circle is equal to or 7. Use this value for the radius to calculate the circumference: C π( r) π(7) 4π 0. The correct answer is B. The standard form of the equation for an ellipse centered at the ( x h) ( y k) point (h, k) is +. In this case, the ellipse is centered at the point (, ). a b Since the value for a is less than b, we also know that this equation has a vertical major axis.. The correct answer is B. First find the domain of the function. In order for h(x) to be a real number, x 0, so x and x 0, which means that x cannot be equal to. The ANSWER KEY - Page 9

20 Recovered Master File (08/08/04 : AM) domain of h(x) is (,). As x runs through (,), range of h(x) is (5, ). x takes on all positive values. The. The correct answer is A. The function F(x) consists of taking the cosine of x, adding 5, and then inverting. Set h(x) cos(x), g(x) x + 5, and f ( x). Calculate [ f g h]( x ) to be sure it x equals the function F(x):. The correct answer is 5.. Multiply the first equation through by : Multiply the second equation through by 4: [ f g h]( x) f ( g( h( x))) f ( g(cos( x))) f (cos( x) + 5) cos( x) + 5 F( x) 5x + 4y 6 (5x + 4y 6) 5x + y 48 6x y 9 4(6x y 9) 4x + y 6 Add the equations together and cancel out the y terms, which leaves a linear equation that can be solved for x: Calculate x + : 5x + y 48 4x y 6 9x x 9 ANSWER KEY - Page 0

21 Recovered Master File (08/08/04 : AM) 8 x The correct answer is even, odd, and neither. Function Even Odd Neither f(x) 4x x g(x) 5sin(θ) x h(x) x x + 5 x For f(x) 4x, you can see on the coordinate plane that this graph is symmetrical about the y-axis, which means that it is an even function. For g(x) 5sin(θ), we know this is an odd function because the sine curve is symmetrical about the origin. For h(x) x x + 5, we know that this is neither odd nor even because the function is shifted and hence will not be symmetrical to the y-axis or the origin. 5. The correct answer is E. x 6 x x 8 6 x (6) x 6 ( 6 ) 4 64 ANSWER KEY - Page

22 Recovered Master File (08/08/04 : AM) 6. The correct answer is D. Plug x 0 into each of the equations to find which one has a y- intercept of : Equation A: Equation B: Equation A: Equation D: Equation E: y x y x y x + x x+ y 7 y sin( x) 0 (0) (0) + sin(0) (0) + (0) The equation given in choice D, y x + 7 has a y-intercept of. 7. The correct answer is D. The notation f(g()) indicates that the value of g() must be determined first, and then that value must be plugged into the equation of f(x). Looking at the table, when x, g(x) 4, and g() 4. Now, plug 4 into the equation f(x): f ( x) x f (4) (4) 6 So, f(g()). 8. The correct answer is A. Use the identity + cot (θ) csc (θ) to solve for csc(θ): + cot ( θ ) csc ( θ) p + ( ) csc ( θ) 6 6 p + csc ( θ) p 6 csc( θ) 9. The correct answer is C. First, calculate the tangent of 0 : ANSWER KEY - Page

23 Recovered Master File (08/08/04 : AM) Plug this value into the equation and solve for y: sin θ tan θ cosθ sin 0 tan0 cos0 5y tan y( ) 0 y The correct answer is C. According to the graph of f(x), f(0). Now look at the graph of g(x) to find the value of g(), which is.. The correct answer is A. The standard form for a parabola with a vertical axis and its vertex at the point (h, k) is (x h) c(y k). An equation for this particular parabola is (x + 6) c(y 8).. The correct answer is B. Plug the equation for h(x) into the equation for g(x): g( h( x)) g(x + 5x 8) (x + 5x 8) + (x + 5x 8) 6 4 (4x + 0x 7x 80x + 64) + (6x + 5x 4) 6 4 4x + 0x x 65x + 4. The correct answer is E. Use the identity sin (θ) +cos (θ) : ANSWER KEY - Page

24 Recovered Master File (08/08/04 : AM) sin ( θ ) + cos ( θ ) z sin ( θ ) + ( ) z sin ( θ ) z sin ( θ ) 4 sin( θ ) ( z)( + z) 4. The correct answer is C. The inverse of a function can be found by reflecting the graph across the line y x, as shown below. Since this graph would not pass the Vertical Line Test, it cannot be a function. 5. The correct answer is E. Use the addition formula for the sine function: sin( α + β ) sin( α)cos( β ) + cos( α)sin( β) sin(50 ) sin( ) (0) () The correct answer is B. Consider each statement. As x moves from 0 to 5, the values of f(x) steadily increase. The statement in choice A is true. According to the table, when f(x) 0, x is equal to. The graph intersects the x-axis at. The statement in choice B is not true about the function. Calculate the slope as x goes from 0 to 5: ANSWER KEY - Page 4

25 Recovered Master File (08/08/04 : AM) y slope x y x 9 ( 6) The slope on this interval is, so the statement in choice C is true. The only statement that is not true about the function is the one given in choice B. 7. The correct answer is B. Use the table to find the value of h(), which is 4. Then use the graph to find the value of g(4), which is. The value of g(h()) is. 8. The correct answer is A. Since Jeffrey worked for 40 hours at $0 per hour, he earned $800. He also made 5% of x dollars, which is equal to 0.5x. In total, Jeffrey earned $ x last week. 9. The correct answer is A. Draw a diagram to represent the situation: Use the trigonometric identity opposite tan( θ ) : adjacent BE tan(45 ) 40 BE 40 BE 40 Also, ANSWER KEY - Page 5

26 Recovered Master File (08/08/04 : AM) Notice that CD AB BE: AB tan(60 ) 40 AB 40 AB 40 CD ( ) 0. The correct answer is 97. First determine the value of g() by plugging into the equation of g(x): Now plug 6 into the equation of f(x) for x: g( x) x 4x 0 g() () 4() 0 (4) f ( x) x 9x + 7 f ( 6) ( 6) 9( 6) The correct answer is 7. feet. Draw a diagram illustrating the situation: Use the trigonometric identity opposite si n( θ ) : hypotenuse ANSWER KEY - Page 6

27 Recovered Master File (08/08/04 : AM) 5 sin(60 ) x x 5 x 0 0 x 7.. The correct answer is E. The domain of the algebraic combination [f + g](x) is defined as dom( f ) dom( g ). According to the graph, the domain of f(x) is all real numbers, or (, ), and the domain of g(x) is all real numbers greater than or equal to, or [, ). The intersection of the domain of f(x) and the domain of g(x) is [, ).. The correct answer is D. Let l equal the length of the rectangle, let w equal the width of the rectangle, and let d equal the length of the diagonal of the rectangle. The length is 4 more than times the width, so l 4 + w. Now use the Pythagorean Theorem to find the length of the diagonal: (4 + w) + w d 6 + 6w + 4w + w d 5w + 6w + 6 d a + b c 5w + 6w + 6 d 4. The correct answer is C. The algebraic combination [f g](x) is defined as f(x) g(x). Plug x 5 into the equation of f(x) to find f(5): f ( x) x 4x 5 f (5) (5) 4(5) 5 (5) Now plug x 5 into the equation for g(x) to find g(5): ANSWER KEY - Page 7

28 Recovered Master File (08/08/04 : AM) Now subtract to find the difference: g( x) g(5) x x 4 + (5) (5) [ f g](5) f (5) g(5) The correct answer is A. Graph the function to observe which y values have corresponding x values: The y values range from to. The range of the function is [, ]. 6. The correct answer is A. Use the double angle formula cos(θ) sin(θ)cos(θ): ANSWER KEY - Page 8

29 Recovered Master File (08/08/04 : AM) cos( θ ) sin( θ)cos( θ) cos( θ ) 4 cos (cos( θ )) cos ( ) 4 θ 4.4 θ 7. The correct answer is B. Use the formula for the volume to solve for h, the height, in terms of r: V π( r) h 0 π( r) h 0 h π ( r) Plug this value into the formula for the surface area: S π ( r) + π( r)( h) 0 π ( r) + π( r)( ) π( r) 60 π ( r) + r 8. The correct answer is and 4. Using the Pythagorean theorem, write an equation for x and y: a + b c ( x) + ( y) (5) 4x + 9y 5 Since the sum of the three sides is, x + y + 5. Solve this equation for either x or y: x + y + 5 x + y 7 x 7 y 7 y x ANSWER KEY - Page 9

30 Recovered Master File (08/08/04 : AM) Plug this value into the first equation: Solve using the quadratic formula: Plug these values into an equation for x: 7 y + y (7 y) + 9y y + 9y + 9y 5 8y 4y ± 764 4(8)(4) y (8) 4 ± ± ± 6 6 4, Calculate the lengths of the legs of the triangle: 7 y x 7 () 4 7 x ANSWER KEY - Page 0

31 Recovered Master File (08/08/04 : AM) x () 4 x y () 4 y 4 The lengths of the legs of the triangle are and 4. f f ( x) 9. The correct answer is A. The algebraic combination ( x) is defined to be, g g( x) provided g(x) 0. According to the tables, f(), and g(). Now divide to find the product: f f () () g g() 40. The correct answer is E. Solve the second equation for q in terms of p: Plug this value into the first equation: p + q 6 p p + q 6 p q 6 p Solve using the quadratic formula: p + pq 4q 4 p + p(6 p) 4(6 p) 4 p + p 4 p 4(6 4 p + 4 p ) 4 p 4 p 6 p + p + 96 p p + 08p p + 08p 58 0 ANSWER KEY - Page

32 Recovered Master File (08/08/04 : AM) 08 ± 664 4( 7)( 58) p ( 7) 08 ± ± , 4.07 Plug these values into an equation for q: q 6 p 6 (.8).44 q 6 (4.07).4 f 4. The correct answer is C. The algebraic combination ( ) x is defined as f ( x ), g g( x) provided g(x) 0. Plug x into the equation for f(x), then multiply through by to find f(): f ( x) x + 6x 7 g() (()) + 6() 7) ((9)) + 8 7) ( ) (9) 87 Now plug x into the equation for g(x), then multiply through by to find g(): g( x) x 4x + g() (()) 4() + ) ((9)) + ) (7 + ) (7) 54 Now divide to find the quotient: ANSWER KEY - Page

33 Recovered Master File (08/08/04 : AM) f f ( x) () g g( x) The correct answer is B. Solve the inequality as if it were an equation. Remember that when dividing or multiplying by a negative number, the inequality sign changes direction: 4 5 p 4 4 5p 4 5 p 0 p 4 Since the problem involves an absolute value, there is a second inequality to solve: 4 5 p 4 5 p 8 p 4. The correct answer is A. The composition function [ f g]( x ) is defined as f(g(x)). According to the table, g () 4. Now look on the table for f(x) to find f(4), which is equal to. Thus: 8 5 [ f g]() f ( g()) f (4) 44. The correct answer is D. Draw a diagram to illustrate the situation: ANSWER KEY - Page

34 Recovered Master File (08/08/04 : AM) Use the trigonometric identity opposite tan( θ ) : adjacent 6 tan( θ ) 8 tan (tan( θ )) tan 4 θ The correct answer is A. Multiply the first equation through by : Multiply the second equation through by : x + 4y 0 (x + 4y 0) 6x y 60 x 5y 6 (x 5y 6) 6x 0y 5 Add the two resulting equations together, canceling out the x terms. Then solve for y: 6x y x 0y 5 y 8 8 y 4 ANSWER KEY - Page 4

35 Recovered Master File (08/08/04 : AM) Plug this value into either of the original equations and solve for x: x + 4y 0 4 x x + 04 x 0 x 46. The correct answer is A. Draw a diagram to illustrate the situation: Use the trigonometric identity opposite tan( θ ) : adjacent 50 tan(0 ) x 50 x x The correct answer is A. Use the addition formula for the sine function: ANSWER KEY - Page 5

36 Recovered Master File (08/08/04 : AM) sin( α + β ) sin( α)cos( β ) + cos( α)sin( β) sin(75 ) sin( ) sin(45 ) cos(0 ) + cos(45 )sin(0 ) The correct answer is B. The result after decreasing by 5% is equal to 0.75x. Decreasing this number by 0% is equivalent to multiplying 0.75x by., which is equal to 0.85x. ANSWER KEY - Page 6

AP Calculus Summer Review Packet

AP Calculus Summer Review Packet AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions

More information

Walt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC

Walt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be

More information

Sec 4.1 Trigonometric Identities Basic Identities. Name: Reciprocal Identities:

Sec 4.1 Trigonometric Identities Basic Identities. Name: Reciprocal Identities: Sec 4. Trigonometric Identities Basic Identities Name: Reciprocal Identities: Quotient Identities: sin csc cos sec csc sin sec cos sin tan cos cos cot sin tan cot cot tan Using the Reciprocal and Quotient

More information

Name Trigonometric Functions 4.2H

Name Trigonometric Functions 4.2H TE-31 Name Trigonometric Functions 4.H Ready, Set, Go! Ready Topic: Even and odd functions The graphs of even and odd functions make it easy to identify the type of function. Even functions have a line

More information

Unit 7: Trigonometry Part 1

Unit 7: Trigonometry Part 1 100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan(

More information

Summer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.

Summer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course. 1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios and Pythagorean Theorem 4. Multiplying and Dividing Rational Expressions

More information

SM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or

SM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length

More information

Summer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.

Summer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course. Summer Review for Students Entering Pre-Calculus with Trigonometry 1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios

More information

Review of Trigonometry

Review of Trigonometry Worksheet 8 Properties of Trigonometric Functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets 8, and The reader should be familiar with the trig ratios,

More information

4-6 Inverse Trigonometric Functions

4-6 Inverse Trigonometric Functions Find the exact value of each expression, if it exists. 29. The inverse property applies, because lies on the interval [ 1, 1]. Therefore, =. 31. The inverse property applies, because lies on the interval

More information

Section 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.

Section 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis. 1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint

More information

Chapter 4: Trigonometry

Chapter 4: Trigonometry Chapter 4: Trigonometry Section 4-1: Radian and Degree Measure INTRODUCTION An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the of the angle, and the position

More information

A lg e b ra II. Trig o n o m e tric F u n c tio

A lg e b ra II. Trig o n o m e tric F u n c tio 1 A lg e b ra II Trig o n o m e tric F u n c tio 2015-12-17 www.njctl.org 2 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector

More information

1. Let be a point on the terminal side of θ. Find the 6 trig functions of θ. (Answers need not be rationalized). b. P 1,3. ( ) c. P 10, 6.

1. Let be a point on the terminal side of θ. Find the 6 trig functions of θ. (Answers need not be rationalized). b. P 1,3. ( ) c. P 10, 6. Q. Right Angle Trigonometry Trigonometry is an integral part of AP calculus. Students must know the basic trig function definitions in terms of opposite, adjacent and hypotenuse as well as the definitions

More information

Mathematics Placement Assessment

Mathematics Placement Assessment Mathematics Placement Assessment Courage, Humility, and Largeness of Heart Oldfields School Thank you for taking the time to complete this form accurately prior to returning this mathematics placement

More information

Trigonometry Summer Assignment

Trigonometry Summer Assignment Name: Trigonometry Summer Assignment Due Date: The beginning of class on September 8, 017. The purpose of this assignment is to have you practice the mathematical skills necessary to be successful in Trigonometry.

More information

Pre-Calculus Summer Assignment

Pre-Calculus Summer Assignment Name: Pre-Calculus Summer Assignment Due Date: The beginning of class on September 8, 017. The purpose of this assignment is to have you practice the mathematical skills necessary to be successful in Pre-Calculus.

More information

SNAP Centre Workshop. Introduction to Trigonometry

SNAP Centre Workshop. Introduction to Trigonometry SNAP Centre Workshop Introduction to Trigonometry 62 Right Triangle Review A right triangle is any triangle that contains a 90 degree angle. There are six pieces of information we can know about a given

More information

You ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46

You ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46 Math 1330 Section 6.2 Section 7.1: Right-Triangle Applications In this section, we ll solve right triangles. In some problems you will be asked to find one or two specific pieces of information, but often

More information

AQA GCSE Further Maths Topic Areas

AQA GCSE Further Maths Topic Areas AQA GCSE Further Maths Topic Areas This document covers all the specific areas of the AQA GCSE Further Maths course, your job is to review all the topic areas, answering the questions if you feel you need

More information

Algebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions

Algebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions 2015-12-17 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 1)

More information

Algebra II Trigonometric Functions

Algebra II Trigonometric Functions Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc

More information

Hiram High School Accelerated Pre-Calculus Summer Assignment

Hiram High School Accelerated Pre-Calculus Summer Assignment Hiram High School Accelerated Pre-Calculus Summer Assignment This is a fast-paced, college-preparatory course that will prepare you to be successful in college math and in AP Calculus. This is a rigorous

More information

Pre-calculus Chapter 4 Part 1 NAME: P.

Pre-calculus Chapter 4 Part 1 NAME: P. Pre-calculus NAME: P. Date Day Lesson Assigned Due 2/12 Tuesday 4.3 Pg. 284: Vocab: 1-3. Ex: 1, 2, 7-13, 27-32, 43, 44, 47 a-c, 57, 58, 63-66 (degrees only), 69, 72, 74, 75, 78, 79, 81, 82, 86, 90, 94,

More information

MA 154 PRACTICE QUESTIONS FOR THE FINAL 11/ The angles with measures listed are all coterminal except: 5π B. A. 4

MA 154 PRACTICE QUESTIONS FOR THE FINAL 11/ The angles with measures listed are all coterminal except: 5π B. A. 4 . If θ is in the second quadrant and sinθ =.6, find cosθ..7.... The angles with measures listed are all coterminal except: E. 6. The radian measure of an angle of is: 7. Use a calculator to find the sec

More information

FUNCTIONS AND MODELS

FUNCTIONS AND MODELS 1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. NEW

More information

Math 1330 Final Exam Review Covers all material covered in class this semester.

Math 1330 Final Exam Review Covers all material covered in class this semester. Math 1330 Final Exam Review Covers all material covered in class this semester. 1. Give an equation that could represent each graph. A. Recall: For other types of polynomials: End Behavior An even-degree

More information

Trigonometric Ratios and Functions

Trigonometric Ratios and Functions Algebra 2/Trig Unit 8 Notes Packet Name: Date: Period: # Trigonometric Ratios and Functions (1) Worksheet (Pythagorean Theorem and Special Right Triangles) (2) Worksheet (Special Right Triangles) (3) Page

More information

DAY 1 - GEOMETRY FLASHBACK

DAY 1 - GEOMETRY FLASHBACK DAY 1 - GEOMETRY FLASHBACK Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse sin θ = opp. hyp. cos θ = adj. hyp. tan θ = opp. adj. Tangent Opposite Adjacent a 2 + b 2 = c 2 csc θ = hyp. opp. sec θ =

More information

Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each.

Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. Math 106/108 Final Exam Page 1 Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. 1. Factor completely. Do not solve. a) 2x

More information

Unit Circle. Project Response Sheet

Unit Circle. Project Response Sheet NAME: PROJECT ACTIVITY: Trigonometry TOPIC Unit Circle GOALS MATERIALS Explore Degree and Radian Measure Explore x- and y- coordinates on the Unit Circle Investigate Odd and Even functions Investigate

More information

4.1: Angles & Angle Measure

4.1: Angles & Angle Measure 4.1: Angles & Angle Measure In Trigonometry, we use degrees to measure angles in triangles. However, degree is not user friendly in many situations (just as % is not user friendly unless we change it into

More information

Ganado Unified School District Pre-Calculus 11 th /12 th Grade

Ganado Unified School District Pre-Calculus 11 th /12 th Grade Ganado Unified School District Pre-Calculus 11 th /12 th Grade PACING Guide SY 2016-2017 Timeline & Resources Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to highlight

More information

Review Notes for the Calculus I/Precalculus Placement Test

Review Notes for the Calculus I/Precalculus Placement Test Review Notes for the Calculus I/Precalculus Placement Test Part 9 -. Degree and radian angle measures a. Relationship between degrees and radians degree 80 radian radian 80 degree Example Convert each

More information

5.1 Angles & Their Measures. Measurement of angle is amount of rotation from initial side to terminal side. radians = 60 degrees

5.1 Angles & Their Measures. Measurement of angle is amount of rotation from initial side to terminal side. radians = 60 degrees .1 Angles & Their Measures An angle is determined by rotating array at its endpoint. Starting side is initial ending side is terminal Endpoint of ray is the vertex of angle. Origin = vertex Standard Position:

More information

Pre Calculus Worksheet: Fundamental Identities Day 1

Pre Calculus Worksheet: Fundamental Identities Day 1 Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy

More information

Trigonometry Review Day 1

Trigonometry Review Day 1 Name Trigonometry Review Day 1 Algebra II Rotations and Angle Terminology II Terminal y I Positive angles rotate in a counterclockwise direction. Reference Ray Negative angles rotate in a clockwise direction.

More information

Solving Trigonometric Equations

Solving Trigonometric Equations OpenStax-CNX module: m49398 1 Solving Trigonometric Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you

More information

Welcome. Please Sign-In

Welcome. Please Sign-In Welcome Please Sign-In Day 1 Session 1 Self-Evaluation Topics to be covered: Equations Systems of Equations Solving Inequalities Absolute Value Equations Equations Equations An equation says two things

More information

Ganado Unified School District Trigonometry/Pre-Calculus 12 th Grade

Ganado Unified School District Trigonometry/Pre-Calculus 12 th Grade Ganado Unified School District Trigonometry/Pre-Calculus 12 th Grade PACING Guide SY 2014-2015 Timeline & Resources Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to

More information

Common Core Standards Addressed in this Resource

Common Core Standards Addressed in this Resource Common Core Standards Addressed in this Resource N-CN.4 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular

More information

1. The Pythagorean Theorem

1. The Pythagorean Theorem . The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be

More information

Trigonometry Curriculum Guide Scranton School District Scranton, PA

Trigonometry Curriculum Guide Scranton School District Scranton, PA Trigonometry Scranton School District Scranton, PA Trigonometry Prerequisite: Algebra II, Geometry, Algebra I Intended Audience: This course is designed for the student who has successfully completed Algebra

More information

Chapter 9: Right Triangle Trigonometry

Chapter 9: Right Triangle Trigonometry Haberman MTH 11 Section I: The Trigonometric Functions Chapter 9: Right Triangle Trigonometry As we studied in Intro to the Trigonometric Functions: Part 1, if we put the same angle in the center of two

More information

Look up partial Decomposition to use for problems #65-67 Do Not solve problems #78,79

Look up partial Decomposition to use for problems #65-67 Do Not solve problems #78,79 Franklin Township Summer Assignment 2017 AP calculus AB Summer assignment Students should use the Mathematics summer assignment to identify subject areas that need attention in preparation for the study

More information

Chapter 4/5 Part 1- Trigonometry in Radians

Chapter 4/5 Part 1- Trigonometry in Radians Chapter 4/5 Part - Trigonometry in Radians Lesson Package MHF4U Chapter 4/5 Part Outline Unit Goal: By the end of this unit, you will be able to demonstrate an understanding of meaning and application

More information

Youngstown State University Trigonometry Final Exam Review (Math 1511)

Youngstown State University Trigonometry Final Exam Review (Math 1511) Youngstown State University Trigonometry Final Exam Review (Math 1511) 1. Convert each angle measure to decimal degree form. (Round your answers to thousandths place). a) 75 54 30" b) 145 18". Convert

More information

5-2 Verifying Trigonometric Identities

5-2 Verifying Trigonometric Identities Verify each identity 1 (sec 1) cos = sin sec (1 cos ) = tan 3 sin sin cos 3 = sin 4 csc cos cot = sin 4 5 = cot Page 1 4 5 = cot 6 tan θ csc tan = cot 7 = cot 8 + = csc Page 8 = csc + 9 + tan = sec 10

More information

5B.4 ~ Calculating Sine, Cosine, Tangent, Cosecant, Secant and Cotangent WB: Pgs :1-10 Pgs : 1-7

5B.4 ~ Calculating Sine, Cosine, Tangent, Cosecant, Secant and Cotangent WB: Pgs :1-10 Pgs : 1-7 SECONDARY 2 HONORS ~ UNIT 5B (Similarity, Right Triangle Trigonometry, and Proof) Assignments from your Student Workbook are labeled WB Those from your hardbound Student Resource Book are labeled RB. Do

More information

Appendix D Trigonometry

Appendix D Trigonometry Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees.

More information

Ganado Unified School District #20 (Pre-Calculus 11th/12th Grade)

Ganado Unified School District #20 (Pre-Calculus 11th/12th Grade) Ganado Unified School District #20 (Pre-Calculus 11th/12th Grade) PACING Guide SY 2018-2019 Timeline & Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to highlight a quantity

More information

College Pre Calculus A Period. Weekly Review Sheet # 1 Assigned: Monday, 9/9/2013 Due: Friday, 9/13/2013

College Pre Calculus A Period. Weekly Review Sheet # 1 Assigned: Monday, 9/9/2013 Due: Friday, 9/13/2013 College Pre Calculus A Name Period Weekly Review Sheet # 1 Assigned: Monday, 9/9/013 Due: Friday, 9/13/013 YOU MUST SHOW ALL WORK FOR EVERY QUESTION IN THE BOX BELOW AND THEN RECORD YOUR ANSWERS ON THE

More information

Algebra II. Slide 1 / 92. Slide 2 / 92. Slide 3 / 92. Trigonometry of the Triangle. Trig Functions

Algebra II. Slide 1 / 92. Slide 2 / 92. Slide 3 / 92. Trigonometry of the Triangle. Trig Functions Slide 1 / 92 Algebra II Slide 2 / 92 Trigonometry of the Triangle 2015-04-21 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 92 Trigonometry of the Right Triangle Inverse

More information

Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using

Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using Ch 13 - RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric

More information

Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: Dear Accelerated Pre-Calculus Student: We are very excited that you have decided to take this course in the upcoming school year! This is a fast-paced, college-preparatory mathematics course that will

More information

PLANE TRIGONOMETRY Exam I September 13, 2007

PLANE TRIGONOMETRY Exam I September 13, 2007 Name Rec. Instr. Rec. Time PLANE TRIGONOMETRY Exam I September 13, 2007 Page 1 Page 2 Page 3 Page 4 TOTAL (10 pts.) (30 pts.) (30 pts.) (30 pts.) (100 pts.) Below you will find 10 problems, each worth

More information

UNIT 5 TRIGONOMETRY Lesson 5.4: Calculating Sine, Cosine, and Tangent. Instruction. Guided Practice 5.4. Example 1

UNIT 5 TRIGONOMETRY Lesson 5.4: Calculating Sine, Cosine, and Tangent. Instruction. Guided Practice 5.4. Example 1 Lesson : Calculating Sine, Cosine, and Tangent Guided Practice Example 1 Leo is building a concrete pathway 150 feet long across a rectangular courtyard, as shown in the following figure. What is the length

More information

PRECALCULUS MATH Trigonometry 9-12

PRECALCULUS MATH Trigonometry 9-12 1. Find angle measurements in degrees and radians based on the unit circle. 1. Students understand the notion of angle and how to measure it, both in degrees and radians. They can convert between degrees

More information

Unit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses.

Unit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Unit 2: Trigonometry This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Pythagorean Theorem Recall that, for any right angled triangle, the square

More information

1.6 Applying Trig Functions to Angles of Rotation

1.6 Applying Trig Functions to Angles of Rotation wwwck1org Chapter 1 Right Triangles and an Introduction to Trigonometry 16 Applying Trig Functions to Angles of Rotation Learning Objectives Find the values of the six trigonometric functions for angles

More information

Moore Catholic High School Math Department

Moore Catholic High School Math Department Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during

More information

Trigonometry. 9.1 Radian and Degree Measure

Trigonometry. 9.1 Radian and Degree Measure Trigonometry 9.1 Radian and Degree Measure Angle Measures I am aware of three ways to measure angles: degrees, radians, and gradians. In all cases, an angle in standard position has its vertex at the origin,

More information

Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics

Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x

More information

Prerequisites for Math 130

Prerequisites for Math 130 Prerequisites for Math 0 The material below represents only some of the basic material with which you should be familiar We will not be reviewing this material You may wish to consult Appendix A in your

More information

Trigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are:

Trigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are: TRIGONOMETRY TRIGONOMETRIC RATIOS If one of the angles of a triangle is 90º (a right angle), the triangle is called a right angled triangle. We indicate the 90º (right) angle by placing a box in its corner.)

More information

CARIBBEAN CORRESPONDENCE SCHOOL

CARIBBEAN CORRESPONDENCE SCHOOL Final Examination CARIBBEAN CORRESPONDENCE SCHOOL Module Name: Groups: Duration: MATHEMATICS Online 3 Hours INSTRUCTIONS TO CANDIDATES 1. This Paper consists of THREE sections. 2. There is one question

More information

Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them.

Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them. Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them. Chapter 9 and 10: Right Triangles and Trigonometric Ratios 1. The hypotenuse of a right

More information

A lg e b ra II. Trig o n o m e try o f th e Tria n g le

A lg e b ra II. Trig o n o m e try o f th e Tria n g le 1 A lg e b ra II Trig o n o m e try o f th e Tria n g le 2015-04-21 www.njctl.org 2 Trig Functions click on the topic to go to that section Trigonometry of the Right Triangle Inverse Trig Functions Problem

More information

Checkpoint 1 Define Trig Functions Solve each right triangle by finding all missing sides and angles, round to four decimal places

Checkpoint 1 Define Trig Functions Solve each right triangle by finding all missing sides and angles, round to four decimal places Checkpoint 1 Define Trig Functions Solve each right triangle by finding all missing sides and angles, round to four decimal places. 1.. B P 10 8 Q R A C. Find the measure of A and the length of side a..

More information

G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S )

G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S ) G r a d e 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 0 S ) Midterm Practice Exam Answer Key G r a d e 0 I n t r o d u c t i o n t o A p p l i e d

More information

You ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46

You ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46 Math 1330 Section 6.2 Section 7.1: Right-Triangle Applications In this section, we ll solve right triangles. In some problems you will be asked to find one or two specific pieces of information, but often

More information

AP Calculus Summer Review Packet School Year. Name

AP Calculus Summer Review Packet School Year. Name AP Calculus Summer Review Packet 016-017 School Year Name Objectives for AP/CP Calculus Summer Packet 016-017 I. Solving Equations & Inequalities (Problems # 1-6) Using the properties of equality Solving

More information

A trigonometric ratio is a,

A trigonometric ratio is a, ALGEBRA II Chapter 13 Notes The word trigonometry is derived from the ancient Greek language and means measurement of triangles. Section 13.1 Right-Triangle Trigonometry Objectives: 1. Find the trigonometric

More information

: Find the values of the six trigonometric functions for θ. Special Right Triangles:

: Find the values of the six trigonometric functions for θ. Special Right Triangles: ALGEBRA 2 CHAPTER 13 NOTES Section 13-1 Right Triangle Trig Understand and use trigonometric relationships of acute angles in triangles. 12.F.TF.3 CC.9- Determine side lengths of right triangles by using

More information

Secondary Math 3- Honors. 7-4 Inverse Trigonometric Functions

Secondary Math 3- Honors. 7-4 Inverse Trigonometric Functions Secondary Math 3- Honors 7-4 Inverse Trigonometric Functions Warm Up Fill in the Unit What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How

More information

Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them.

Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them. Semester 2 Review Problems will be sectioned by chapters. The chapters will be in the order by which we covered them. Chapter 9 and 10: Right Triangles and Trigonometric Ratios 1. The hypotenuse of a right

More information

Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231

Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231 1 Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231 What is Trigonometry? 2 It is defined as the study of triangles and the relationships between their sides and the angles between these sides.

More information

Warm-Up: Final Review #1. A rectangular pen is made from 80 feet of fencing. What is the maximum area the pen can be?

Warm-Up: Final Review #1. A rectangular pen is made from 80 feet of fencing. What is the maximum area the pen can be? Warm-Up: Final Review #1 A rectangular pen is made from 80 feet of fencing. What is the maximum area the pen can be? Warm-Up: Final Review #2 1) Find distance (-2, 4) (6, -3) 2) Find roots y = x 4-6x 2

More information

Math 144 Activity #2 Right Triangle Trig and the Unit Circle

Math 144 Activity #2 Right Triangle Trig and the Unit Circle 1 p 1 Right Triangle Trigonometry Math 1 Activity #2 Right Triangle Trig and the Unit Circle We use right triangles to study trigonometry. In right triangles, we have found many relationships between the

More information

Exam 2 Review. 2. What the difference is between an equation and an expression?

Exam 2 Review. 2. What the difference is between an equation and an expression? Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? 2. What the difference is between an equation and an expression? 3. How to tell if an equation is linear? 4. How

More information

sin30 = sin60 = cos30 = cos60 = tan30 = tan60 =

sin30 = sin60 = cos30 = cos60 = tan30 = tan60 = Precalculus Notes Trig-Day 1 x Right Triangle 5 How do we find the hypotenuse? 1 sinθ = cosθ = tanθ = Reciprocals: Hint: Every function pair has a co in it. sinθ = cscθ = sinθ = cscθ = cosθ = secθ = cosθ

More information

MATHEMATICS FOR ENGINEERING TRIGONOMETRY

MATHEMATICS FOR ENGINEERING TRIGONOMETRY MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL SOME MORE RULES OF TRIGONOMETRY This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves

More information

Calculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier

Calculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier Calculus I Review Handout 1.3 Introduction to Calculus - Limits by Kevin M. Chevalier We are now going to dive into Calculus I as we take a look at the it process. While precalculus covered more static

More information

Graphing Trigonometric Functions: Day 1

Graphing Trigonometric Functions: Day 1 Graphing Trigonometric Functions: Day 1 Pre-Calculus 1. Graph the six parent trigonometric functions.. Apply scale changes to the six parent trigonometric functions. Complete the worksheet Exploration:

More information

COMPASS/ESL Sample Test Questions A Guide for Students and Parents Mathematics

COMPASS/ESL Sample Test Questions A Guide for Students and Parents Mathematics COMPASS/ESL Sample Test Questions A Guide for Students and Parents Mathematics College Algebra Geometry Trigonometry An ACT Program for Educational Planning Note to Students Welcome to the COMPASS Sample

More information

Triangle Trigonometry

Triangle Trigonometry Honors Finite/Brief: Trigonometry review notes packet Triangle Trigonometry Right Triangles All triangles (including non-right triangles) Law of Sines: a b c sin A sin B sin C Law of Cosines: a b c bccos

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Precalculus CP Final Exam Review - 01 Name Date: / / MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle in degrees to radians. Express

More information

TABLE 2: Mathematics College Readiness Standards for Score Range 13 15

TABLE 2: Mathematics College Readiness Standards for Score Range 13 15 TABLE 2: Mathematics College Readiness Standards for Score Range 13 15 Perform one-operation computation with whole numbers and decimals Solve problems in one or two steps using whole numbers Perform common

More information

Unit 2 Intro to Angles and Trigonometry

Unit 2 Intro to Angles and Trigonometry HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of

More information

UPCAT Reviewer Booklet

UPCAT Reviewer Booklet UPCAT Reviewer Booklet I. Linear Equations y = y-value at a certain point in the graph x = x-value at a certain point in the graph b = a constant m = the slope of the line Section 1 Mathematics Linear

More information

Albertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.

Albertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school. Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the

More information

This unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA.

This unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA. Angular Rotations This unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA. sin x = opposite hypotenuse cosx =

More information

Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8)

Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Standards: Geom 19.0, Geom 20.0, Trig 7.0, Trig 8.0, Trig 12.0 Segerstrom High School -- Math Analysis Honors Name: Period:

More information

2.0 Trigonometry Review Date: Pythagorean Theorem: where c is always the.

2.0 Trigonometry Review Date: Pythagorean Theorem: where c is always the. 2.0 Trigonometry Review Date: Key Ideas: The three angles in a triangle sum to. Pythagorean Theorem: where c is always the. In trigonometry problems, all vertices (corners or angles) of the triangle are

More information

Santiago AP Calculus AB/BC Summer Assignment 2018 AB: complete problems 1 64, BC: complete problems 1 73

Santiago AP Calculus AB/BC Summer Assignment 2018 AB: complete problems 1 64, BC: complete problems 1 73 Santiago AP Calculus AB/BC Summer Assignment 2018 AB: complete problems 1 64, BC: complete problems 1 73 AP Calculus is a rigorous college level math course. It will be necessary to do some preparatory

More information

FORMULAS to UNDERSTAND & MEMORIZE

FORMULAS to UNDERSTAND & MEMORIZE 1 of 6 FORMULAS to UNDERSTAND & MEMORIZE Now we come to the part where you need to just bear down and memorize. To make the process a bit simpler, I am providing all of the key info that they re going

More information

High School MATHEMATICS Trigonometry

High School MATHEMATICS Trigonometry High School MATHEMATICS Trigonometry Curriculum Curriculum Map USD 457 Math Framework Performance/Open-Ended Response Assessments o First Semester (Tasks 1-3) o Second Semester (Tasks 1-4) Available on

More information

Mastery. PRECALCULUS Student Learning Targets

Mastery. PRECALCULUS Student Learning Targets PRECALCULUS Student Learning Targets Big Idea: Sequences and Series 1. I can describe a sequence as a function where the domain is the set of natural numbers. Connections (Pictures, Vocabulary, Definitions,

More information

1. (10 pts.) Find and simplify the difference quotient, h 0for the given function

1. (10 pts.) Find and simplify the difference quotient, h 0for the given function MATH 1113/ FALL 016 FINAL EXAM Section: Grade: Name: Instructor: f ( x h) f ( x) 1. (10 pts.) Find and simplify the difference quotient, h 0for the given function h f ( x) x 5. (10 pts.) The graph of the

More information