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

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1 Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector Unit Circle Graphing Trigonometric Identities

2 Slide 4 / 162 Radians & Degrees and Co-Terminal Angles Return to Table of Contents A few definitions: A central angle of a circle is an angle whose vertex is the center of the circle. An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle. Slide 5 / 162 intercepted arc central angle Radians and Degrees Slide 6 / 162 One radian is the measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. There are, or a little more than 6, radians in a circle. Click on the circle for an animated view of radians.

3 Converting from Degrees to Radians Slide 7 / 162 There are 360 in a circle. Therefore 360 = 2 radians 2 1 = 360 = 180 radians Use this conversion factor to covert degrees to radians. Example: Convert 50 and 90 to radians. 50 = 5 radians = radians Converting from Radians to Degrees Slide 8 / radians = radian = = 180 degrees 2 Use this conversion factor to covert radians to degrees. Example: Convert and to radians = = 180 Converting between Radians and Degrees Slide 9 / 162 Convert degrees to radians

4 Converting between Radians and Degrees Slide 10 / 162 Convert radians to degrees radians radians radians Slide 11 / 162 Slide 12 / 162

5 Slide 13 / 162 Slide 14 / Convert radians to degrees: Angles Slide 15 / 162 Terminal side Initial side Terminal side Initial side Angle Angle in standard position An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side. When, on the coordinate plane, the vertex of the angle is the origin and the initial side is the positive x-axis, the angle is in standard position.

6 Positive Angle - terminal side rotates in a counterclockwise direction Negative Angle - terminal side rotates in a clockwise direction Slide 16 / 162 α = - 37 Drawing angles in standard position Slide 17 / Each quadrant is 90, and 310 is 40 more than 270, so the terminal side is 40 past the negative y-axis. 500 is 140 more than 360, so the angle makes a complete revolution counterclockwise and then another 140. Coterminal Angles Slide 18 / 162 Angles that have the same terminating side are coterminal. To find coterminal angles add or subtract multiples of 360 for degrees and 2 for radians. Example: Find one positive and one negative angle that are terminal with = =

7 5 Which angles are coterminal with 40? (Select all that are correct.) A 320 B -320 C 400 D -400 Slide 19 / Which graph represents 425? Slide 20 / 162 A B C D 7 Which graph represents? Slide 21 / 162 A B C D

8 8 Which angle is NOT coterminal with -55? Slide 22 / 162 A 305 B 665 C -415 D Which angle is coterminal with? Slide 23 / 162 A B C D Slide 24 / 162 Arc Length & Area of a Sector Return to Table of Contents

9 Slide 25 / 162 Arc length and the area of a sector (Measured in radians) r arc length s sector Arc length: s = r Area of sector: A = How do these formulas relate to the area and the circumference of a circle? Who is getting more pie? Who is getting more of the crust at the outer edge? Slide 26 / Emily's slice is cut from a 9 inch pie. Chester's slice is cut from an 8 inch pie. (Assume both pies are the same height.) (Try to work this out in your groups. The solution is on the next slide) Slide 27 / click The top of Emily's piece has an area of click The top of Chester's piece has an area of Emily's crust has a length of Chester's crust has a length of

10 10 What is the top surface area of this slice of pizza from an 18-inch pie? Slide 28 / What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? Slide 29 / If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth? Slide 30 / 162

11 13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region? Slide 31 / in 8 inches Slide 32 / 162 Unit Circle Return to Table of Contents The Unit Circle Slide 33 / 162 The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant I: x and y are both positive (-1,0) Quadrant III: x and y are both negative (0,-1) (1,0) Quadrant IV: x is positive and y is negative

12 The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures. (-1,0) (0,-1) (0,1) 1 θ a (a,b) b (1,0) In this triangle, b sin#= 1 = b a cos# = 1 = a so the coordinates of (a,b) are also (cos#, sin#) Slide 34 / 162 For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point. In this example, the terminal point is in Quadrant IV If we look at the triangle, we can see that sin(-55 ) = 0.82 cos(-55 ) = 0.57 EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis. Slide 35 / 162 For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ). Slide 36 / 162 Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:

13 What are the coordinates of point C? Slide 37 / 162 In this example, we know the angle. Using a calculator, we find that cos and sin 44.69, so the coordinates of C are approximately (0.72, 0.69). 1 Note that ! The Tangent Function Recall SOH-CAH-TOA Slide 38 / 162 sin # = opp hyp cos # = adj hyp opp tan # = adj opposite side hypotenuse # adjacent side It is also true that tan = sin # cos. # Why? opp hyp = opp hyp = opp adj hyp adj adj = tan # hyp Angles in the Unit Circle Slide 39 / 162 Example: Given a terminal point on the unit circle (- ). Find the value of cos, sin and tan of the angle. Solution: Let the angle be. x = cos, so cos =. y = sin, so sin =. tan = = = = (Shortcut: Just cross out the 41's in the complex fraction.)

14 Example: Given a terminal point csc#. Note the "hidden" Pythagorean Triple, 8, 15, 17)., find #, tan# and To find #, use sin -1 or cos -1 : sin -1 ( ) = # # # 28.1 tan# = sin#/ cos# tan # = csc# = 1/ sin# csc # = Slide 40 / 162 Example: Find the x-value of point A, θ and the tan θ. For every point on the circle, Slide 41 / (, - 13 ) θ A Since x is in quadrant III, x = sin -1 5 (- 13 ) -22.3, BUT θ is in quadrant III, so θ = = (notice how and have the same sine) sin θ tan θ = cos θ = = 5 12 Example: Given the terminal point of ( -5 / 13, -12 / 13). Find sin x, cos x, and tan x. Slide 42 / 162

15 14 What is tan θ? 3 (- 5, ) Slide 43 / 162 A θ B C D 15 What is sin θ? 3 (- 5, ) Slide 44 / 162 A θ B C D 16 What is θ (give your answer to the nearest degree)? Slide 45 / (- 5, ) θ

16 17 Given the terminal point, find tan x. Slide 46 / 162 Slide 47 / 162 Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45. A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values. Slide 48 / 162 Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two triangles)

17 Special Right Triangles Slide 49 / 162 (see Triangle Trig Review unit for more detail on this topic) Special Triangles and the Unit Circle Slide 50 / 162 (-, ) (, ) Multiples of 45 angles have sin and cos of ±, depending on the quadrant. 30 o 45 o 60 o 60 o Slide 51 / o 30 o 30 o 30 o 45 o 60 o 60 o 45 o

18 Drag the degree and radian angle measures to the angles of the circle: # 5# # 3# 7# 3# 0 # 2# Slide 52 / Fill in the coordinates of x and y for each point on the unit circle: Slide 53 / 162 (, ) 3# 4 (, ) # 2 # 4 (, ) (, ) # (, ) 5# 4 3# 2 7# 4 2# 0 (, ) (, ) (, ) Special Triangles and the Unit Circle Slide 54 / 162 (, ) 1 (, ) Angles that are multiples of 30 have sin and cos of ± and ±.

19 Drag the degree and radian angle measures to the angles of the circle: 5# # # # 3# 0 2# 4# # 2# 7# 11# 5# Slide 55 / Drag in the coordinates of x and y for each point on the unit circle: Slide 56 / 162 (, ) (, ) (, ) 2# 3 # 5# 6 7# (, ) 6 4# 3 (, ) # 2 (, ) (, ) # 3 (, ) # 6 2# 0 (, ) 11# (, ) 6 3# 5# 2 3 (, ) (, ) Special Angles in Degrees Slide 57 / 162

20 Radian Values of Special Angles Slide 58 / 162 Exact Values of Special Angles Slide 59 / 162 Put it all together... Slide 60 / 162

21 Exact values of special angles Slide 61 / 162 Complete the table below: Degrees Radians sin θ cos θ tan θ Slide 62 / 162 Slide 63 / 162

22 Slide 64 / 162 If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values. Example: If tan =, and sin < 0, find sin, cos and the value of. Slide 65 / 162 Solution: Since tan is positive and sin is negative, the terminal side of must be in Quadrant III. Draw a right triangle in Quadrant III. Use the Pythagorean Theorem to find the length of the hypotenuse: opp -3 adj hyp (Continued on next slide) Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values. opp -3 adj hyp Slide 66 / 162 sin = = cos = = Use any inverse trig function to find the angle. tan-1( ) Because the angle is in QIII, we need to add = 216.7, so 217.

23 Slide 67 / 162 Slide 68 / 162 Slide 69 / 162

24 Slide 70 / Which functions are positive in the second quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Slide 71 / Which functions are positive in the fourth quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Slide 72 / Which functions are positive in the third quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x

25 Slide 73 / 162 Graphing Trig Functions Return to Table of Contents If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions: Slide 74 / 162 (Once the webpage opens, click on Download) Graphing the Sine Function, y = sin x Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of, and the corresponding values of y or sin x. Slide 75 / 162 (Remember, is just a bit more than 3.) Since the values are based on a circle, values will repeat.

26 Notice that the graph of y = sin x increases from 0 to 1, then decreases back to 0 and then to -1 and then goes back up to 0, as x increases from 0 to 2. Slide 76 / 162 Graphing the Cosine Curve Slide 77 / 162 Make a table of values just as we did for sin. We could use any interval, but are choosing from 0 to 2. Since the values are based on a circle, values will repeat. Notice that the graph of y = cos x starts at 1, decreases to -1 and then goes back up to 1 as x increases from 0 to 2. Slide 78 / 162

27 Compare the graphs: Slide 79 / 162 y = sin x y = cos x How are they similar and how are they different? Characteristics of y = sin x and y = cos x Slide 80 / 162 range: -1 y 1 amplitude = 1 period = 2 Domain: set of real numbers (x can be anything) Range: -1 y 1 Amplitude: one-half the distance from the minimum of the graph to the maximum or 1. The functions are periodic - the pattern repeats every 2 units. Predict, Explore, Confirm Slide 81 / Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?

28 y = a sin x or y = a cos x Slide 82 / 162 Amplitude is a positive number that represents one-half the difference between the minimum and the maximum values, or the distance from the midline to the maximum. Consider the graphs of y = sin x What do you notice about these y = 2sin x graphs? What does the value of y = sin x "a" do to the graph? Slide 83 / 162 y = 2sin x y = sin x y = sin x Name the amplitude of each graph. As shown in the graph below, the graph of y = -3cos xis a reflection over the x-axis of the graph of y = 3cos x. What is the amplitude of each function? y = 3cos x y = -3cos x Slide 84 / 162 The domain of each function is the set of real numbers and the range is {x -3 x 3}.

29 Sketch each graph on the interval from 0 to 2 : Slide 85 / 162 y = 4cos x y = -.25 sin x Slide 86 / What is the amplitude of y = 3cos x? Slide 87 / What is the amplitude of y = 0.25cos x?

30 Slide 88 / What is the amplitude of y = -sin x? 28 What is the range of the function y = 2sin x? Slide 89 / 162 A All real numbers B -2 < x < 2 C 0 x 2 D -2 x 2 29 What is the domain of y = -3cos x? Slide 90 / 162 A All real numbers B -3 < x < 3 C 0 x 3 D -3 x 3

31 30 Which graph represents the function y = -2sin x? Slide 91 / 162 A B C D 31 What is the amplitude of the graph below? Slide 92 / 162 Predict, Explore, Confirm Slide 93 / Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?

32 Slide 94 / 162 A periodic function is one that repeats its values at regular intervals. One complete repetition of the pattern is called a cycle. The period is the length of one complete cycle. The trig functions are periodic functions. The basic sine and cosine curves have a period of 2, meaning that the graph completes one complete cycle in 2 units. y = sin bx or y = cos bx Slide 95 / 162 Consider the graphs of y = cos xand y = cos 2x. y = cos x one cycle y = cos 2x Notice that the graph of y = cos 2xcompletes one cycle twice as fast, or in units. y = cos x completes 1 cycle in 2#. So the period is 2π. Slide 96 / 162 y = cos 2x completes 2 cycles in 2# or 1 cycle in #. The period is #. y = cos 0.5x completes a cycle in 4#. The period is 4#.

33 Slide 97 / 162 The period for y = cos bx or y = sin bx is P = 2 b y = cos x b = 1 2 P = 1 = 2 y = cos 2x b = 2 P = 2 2 = y = cos 0.5x b = 0.5 P = = 4 Slide 98 / What is the period of A B C D 33 What is the period of Slide 99 / 162 A B C D

34 Slide 100 / What is the period of A B C D Sketch the graph of each function from x = 0 to x = 2. Slide 101 / 162 y = 2cos 3x y = cos x y = sin 2x y = -2cos 2x 35 What is the period of the graph below? Slide 102 / 162 A B 2 C 3 2 D 2

35 Slide 103 / What is the period of the graph shown? A B C D Slide 104 / What is the equation of this function? A B C D y = sin 3x y = cos 3x y = 3cos x y = 3sin x Predict, Explore, Confirm Slide 105 / Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?

36 Translating Sine and Cosine Functions Slide 106 / 162 Trig functions can be translated in the same way as any other function. The horizontal shift is called a phase shift. What are your conclusions from the graphing calculator activity? Drag each equation to the matching graph Slide 107 / 162 Horizontal or phase shift y = cos x y = cos (x + ) 2 Vertical shift y = sin x y = sin x + 2 k Slide 108 / 162

37 Slide 109 / 162 Consider the graphs of Slide 110 / 162 and (which is which?) In order to determine the phase shift, the coefficient of x must be factored out. In shift is. In, the 2 is factored out. The phase, when the 2 is factored out, we get. The phase shift is. Slide 111 / 162 Another way to determine the phase shift is to set the quantity inside the parenthesis equal to 0 and solve for the variable. Example: Set Solve for x: So, the phase shift is 2.

38 Slide 112 / 162 Slide 113 / 162 Vertical Shift y= sin (x) + k or y= cos (x) + k The k moves the graph up or down. The graph below is of the equation y = 2 sin (3x). The midline of this graph is the horizontal line y = 0. Sketch the graph of y = 2 sin (3x) + 1. Slide 114 / 162

39 Slide 115 / What is the vertical shift in Slide 116 / What is the vertical shift in Slide 117 / What is the vertical shift in

40 Slide 118 / 162 Graphing a Sine or Cosine Function: Slide 119 / 162 Step 1: Identify the amplitude, period, phase shift and vertical shift. Step 2: Draw the midline (y = k) Step 3: Find 5 key points - maximums, minimums and points on the midline Step 4: Draw the graph through the 5 points. Example: Slide 120 / 162 Step 1: Amplitude: -1 = 1 Period: Phase Shift: Vertical Shift: 2 (up 2)

41 Step 2: Draw the midline y = 2 Step 3: Find the 5 key points Slide 121 / 162 Note: for x, adding the cycle, 3 by 4. comes from dividing For y, adding and subtracting 1 comes from the amplitude. Slide 122 / 162 Step 4: Graph You try: Slide 123 / 162

42 Slide 124 / 162 Slide 125 / 162 Slide 126 / 162

43 Slide 127 / 162 Slide 128 / What is the amplitude of this cosine graph? Slide 129 / What is the period of this cosine graph? (use 3.14 for pi)

44 Slide 130 / What is the vertical shift of this cosine graph? Slide 131 / Which of the following of the following is an equation for the graph? A B C D The equation y = 4.2cos (π/6(x - 1)) can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, Sketch the graph of this equation. What is the average temperature in June? Slide 132 / 162

45 Graphing the Tangent Function Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of, and the corresponding values of y or tan x. Slide 133 / 162 Notice that the tangent function is not defined for values of x where cos x = 0, or starting at and every units in either direction. Slide 134 / 162 This is shown on the graph by the vertical lines, or asymptotes at these x values. The period of the function is units, because there is one complete cycle from to. As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity. Example: Sketch the graph of y = tan (x + ) + 2 Slide 135 / 162 Asymptotes will be at 0,, 2, etc. The midline will be at y = 2.

46 53 Which graph represents y = -tan x? Slide 136 / 162 A B C D Slide 137 / 162 Trigonometric Identities Return to Table of Contents Key Ideas An identity is a mathematical equation that is true for all defined values of the variable. A trigonometric identity is an identity that contains one or more trig ratios. By contrast, a conditional equation is one that is only true for a limited set of values. By learning trig identities, we will be able to replace complicated expressions with simpler ones to solve and verify more difficult equations and identities. Slide 138 / 162

47 Drag each equation into the correct box: Slide 139 / 162 3x + 4 = 3x + 4 3x + 4 = 9 5x - 7y = -(7y - 5x) 2x 5 =x 3 sin # + cos # = 1 tan θ cot θ =1 2(x-1) = 2x - 2 (x + 3) 2 = x sin 4x = 4sin x Identities Conditional Equations Basic Trig Identities Slide 140 / 162 Reciprocal Identities csc # = 1 sin # sin # = 1 csc # sec # = 1 cos # cos # = 1 sec # cot # = 1 tan # tan # = 1 cot # Tangent Identity tan # = sin # cos # Cotangent Identity cot # = cos # sin # Slide 141 / 162 By using the basic identities we can change an expression into an equivalent expression. Also, all of the rules of addition, subtraction, multiplication and division that we learned to solve equations and manipulate expressions can be applied to trig expressions and equations.

48 Slide 142 / 162 Algebraic example Trig example (x - y)(x + y) = x 2 - y 2 (1 - cos #)(1 + cos #) = 1 - cos 2 # Pythagorean Identities Slide 143 / 162 Recall the unit circle, x 2 + y 2 = 1. (-1,0) (0,1) 1 (0,-1) (cos #,sin #) (1,0) For any point (x, y) on the circle, its coordinates are (cos #, sin #). Therefore, (cos #) 2 + (sin #) 2 = 1 2, which is usually written as cos 2 θ + sin 2 θ = 1 Slide 144 / 162 Pythagorean Identities How do we transform the first identity, which is derived from the unit circle, to the other two?

49 Alternative Forms of Identities Slide 145 / 162 Since we know that = 8, we also know that 8-5 = 3 and 8-3 = 5. In elementary school we call these equivalent equations "fact families". Similarly, if cos 2 θ + sin 2 θ = 1, it follows that 1 - cos 2 θ = sin 2 θ and 1 - sin 2 θ = cos 2 θ. More Alternative Forms Slide 146 / 162 Another fact family tells that since follows that 4 5 = = 4, it 1 Since sec θ = cos θ, then sec θ cos θ = 1 (multiply both sides of the first equation by cos #). Simplifying Trig Expressions Slide 147 / 162 Example 1: Simplify csc θ cos θ tan θ. Rewrite each trig ratio in terms of cos and sin: 1 sin # sin θ cos θ cos # = 1 Example 2: Simplify csc 2 θ(1 - cos 2 θ). 1 sin 2 θ (sin2 θ) = 1 (When multiplying fractions, it is often easier to reduce or cancel before you multiply.)

50 Verifying an Identity Transform one side of the identity to be the same as the other side Slide 148 / 162 Example 1: Verify sin # cot # = cos # sin # cos # = cos # sin # Example 2: Verify cos θ csc θ tan θ = 1 cos # 1 sin # = 1 sin # cos # Simplify: Slide 149 / 162 Simplify: Slide 150 / 162

51 Simplify: Slide 151 / 162 Slide 152 / 162 Slide 153 / 162 Simplify:

52 Verify: Slide 154 / 162 Slide 155 / 162 Verify: Slide 156 / 162

53 Slide 157 / 162 Slide 158 / Which equation is NOT an identity? Slide 159 / 162 A sin 2 x= 1 - cos 2 x B 2 cot x = 2cos x sin x C tan 2 x = sec 2 x - 1 D sin 2 x = cos 2 x - 1

54 55 The following expression can be simplified to which choice? Slide 160 / 162 A B C D 56 The following expression can be simplified to which choice? Slide 161 / 162 A B C D 57 The following expression can be simplified to which choice? Slide 162 / 162 A B C D

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