Math 1330 Test 3 Review Sections 4.1 4.3, 5.1a, 5. 5.4; Know all formulas, properties, graphs, etc! 1. Similar to a Free Response! Triangle ABC has right angle C, with AB = 9 and AC = 4. a. Draw and label the triangle, and find the length of the missing side. b. State the six trigonometric functions of angle A. 4. Suppose that α is an acute angle in a right triangle and that tan( α ) =. Draw the 7 triangle and find the missing side. Then find the rest of the trigonometric functions of α. Math 1330 Test 3 Review 1
3. Convert: a. 19π to degrees. b. o 40 to radians 0 4. A sector of a circle has central angle θ = π/3 and area 49π/6 ft. Find the radius of 1 the circle. Recall: A = r θ 5. If the central angle θ defining the sector is given in degrees, then the area of the sector θ can be found using the formula: A = π r Use the formula to find the area of the 360 sector with θ = 00 o and r = 6 cm. 6. A car has wheels with a 18 cm radius. If each wheel s rate of turn is 6 revolutions per second, how fast is the car moving in units of cm/sec? s θ s rθ Recall: θ = so s = rθ. Angular velocity is: ω = ; Linear velocity is: v = = r t t t Math 1330 Test 3 Review
7. Find three angles, two positive and one negative, that are coterminal with the given angle: -10π 9 Recall: x o + k360 o = x + π k 8. Evaluate. 7π a. sin 4 b. 7π cot 6 o c. sec ( 70 ) d. cos ( 135 o ) Math 1330 Test 3 Review 3
Recall: sin( t + kπ) = sin(t), csc( t + πk) = csc(t) cos( t + kπ) = cos(t), sec( t + πk) = sec(t) tan( t + kπ) = tan(t), cot( t + kπ) = cot(t), for all real numbers t and all integers k. 9. Simplify. 8π 31π 3sin + tan 3 6 Recall: RESTRICED DOMAINS QUADRANTS 1 AND 4 π π Inverse sine:,, Inverse cosecant: π π π π,0 0,, Inverse tangent:, QUADRANTS 1 AND Inverse cosine: [ 0,π ], Inverse secant: π, π, π 0, Inverse cotangent: ( 0,π ) 10. Evaluate. 1 a. 3 sin 1 1 b. cos ( 0) 1 c. tan ( 1) Math 1330 Test 3 Review 4
1 8 11. Find sec sin. 11 1. Find the period. 13. Similar to a Free Response! Given the following sine curve and the fact that the point (π, 5) is on the graph, what is the equation in terms of sine that produces this graph? Find A by noting the amplitude: A Find any Vertical Shift, D: It ll be half-way between the maximum and the minimum values. Find B by using the period formula for sine: π B Find any phase shift, C: Any other transformations? Math 1330 Test 3 Review 5
14. Give an equation of the form f (x) = A cos(b x - C) + D which could be used to represent the given graph. (Note: C or D may be zero.) Choose a cycle. Amplitude: A = M m = Vertical Shift, D: It ll be half-way between the maximum and the minimum values. Use the period to find B: Recall the period formula π = B Compare your chosen cycle to the basic cycle of cosine, any other transformations? a) b) c) d) e) 15. Give an equation of the form f (x) = A tan(b x - C) + D which could be used to represent the given graph. (Note: C or D may be zero.) Choose a cycle. Find A by using the y values of the second and fourth points and plug big small them into A = = Vertical Shift, D: It ll be half-way between the big y-value and the small y-value. π Use the period to find B: Recall the period formula = B Compare your chosen cycle with the basic one cycle of tangent. Any other transformations? a) b) c) d) e) Math 1330 Test 3 Review 6
16. Find the period and phase shift for the following function 1 f ( x) = 5cos πx π 3. 4 Period formula: π B 17. Find the phase shift for the following function 1 f ( x) = 8sin π x + π + 4. 3 Phase shift formula, assuming B was not factored out. C B Now do it by factoring B out, it s the same phase shift. 1 f ( x) = 8sin π x + π + 4 = 3 1 18. List all x-intercepts for y = 5sin 4x + π on the interval [-π/6, π/ ]. 3 Recall: Use the graph above to recall the ORIGINAL x-intercepts of sine. Then set the inside equal to one, solve for x and check the answers are in the interval given. Math 1330 Test 3 Review 7
19. Which of these is an equation of one of the asymptotes of the following function? I. f ( x) = 7cot(4x) 1 Recall: Use the graph above to recall the ORIGINAL asymptotes of cotangent. Then set the inside equal to one and solve for x. A. 1 B. 4 1 1 1 C. π D. π 8 4 8 π π II. g ( x ) = sec x + 4 Recall: Use the graph above to recall the ORIGINAL asymptotes of secant. Then set the inside equal to one and solve for x. A. π B. 4 π C. 1 D. 3 0. Write a sine function with a positive vertical dilation, given the amplitude is 5, the phase shift is 4 to the left, the vertical shift is 3 down, and the period is. 1. : The number of hours of daylight in Boston is given by π f ( x) = 3sin ( x 79) + 1, where x is the number of days after January 1. Find the 365 amplitude and the period Math 1330 Test 3 Review 8
. Similar to a Free Response! Sketch the graph of f ( x) = 5sin(x) for one period. Include any vertical/horizontal asymptotes if any. MAKE SURE PARTS a, b AND c BELOW ARE LABELED ON THE GRAPH ITSLEF!! a. Label the x and y intercepts (if any) with ordered pairs. b. Label the maximum value(s) (if any) with an ordered pair. c. Label the minimum value(s) (if any) with an ordered pair. d. What is the domain of f ( x) = 5sin(x)? e. What is the range of f ( x) = 5sin(x)? Math 1330 Test 3 Review 9
3. Similar to a Free Response! Sketch the graph of g ( x) = 7 cos(5x) for one period. Include any vertical/horizontal asymptotes if any. MAKE SURE PARTS a, b AND c BELOW ARE LABELED ON THE GRAPH ITSLEF!! a. Label the x and y intercepts (if any) with ordered pairs. b. Label the maximum value(s) (if any) with an ordered pair. c. Label the minimum value(s) (if any) with an ordered pair. d. What is the domain of g ( x) = 7 cos(5x)? e. What is the range of g ( x) = 7 cos(5x)? Math 1330 Test 3 Review 10
f ( x) = cos( x) f ( x) = csc( x) f ( x) = tan( x) Math 1330 Test 3 Review 11