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 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 Unit Circle Graphing Trigonometric Identities 3

Radians & Degrees and Co-Terminal Angles Return to Table of Contents 4

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. central angle intercepted arc 5

Radians and Degrees 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. 6

Converting from Degrees to Radians 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 180 18 90 = radians 180 2 7

Converting from Radians to Degrees 2 radians = 360 1 radian 360 = = 180 2 degrees Use this conversion factor to covert radians to degrees. Example: Convert and to radians 4 4 180 = 45 180 = 180 8

Converting between Radians and Degrees Convert degrees to radians Answer 9

Converting between Radians and Degrees Convert radians to degrees radians radians Answer radians 10

1 Convert degrees to radians: A B C Answer D 11

2 Convert degrees to radians: A B C Answer D 12

3 Convert radians to degrees: Answer 13

4 Convert radians to degrees: Answer 14

Angles 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. 15

Positive Angle - terminal side rotates in a counterclockwise direction Negative Angle - terminal side rotates in a clockwise direction α = - 37 16

Drawing angles in standard position 310 500 40 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. 17

Coterminal Angles 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 75. 75 + 360 = 435 75-360 = -285 435-285 75 18

5 Which angles are coterminal with 40? (Select all that are correct.) A 320 B -320 C 400 D -400 Answer 19

6 Which graph represents 425? A B Answer C D 20

7 Which graph represents? A B Answer C D 21

8 Which angle is NOT coterminal with -55? A 305 B 665 C -415 D -305 Answer 22

9 Which angle is coterminal with? A B Answer C D 23

Arc Length & Area of a Sector Return to Table of Contents 24

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? 25

Who is getting more pie? Who is getting more of the crust at the outer edge? 40 45 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) 26

40 45 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 27

10 What is the top surface area of this slice of pizza from an 18-inch pie? 45 Answer 28

11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? 45 Answer 29

12 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? Answer 30

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? 4 in 8 inches Answer 31

Unit Circle Return to Table of Contents 32

The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. 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 33

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, sinθ= b 1 cosθ = = b a 1 = a so the coordinates of (a,b) are also (cosθ, sinθ) For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point. 34

In this example, the terminal point is in Quadrant IV. If we look at the triangle, we can see that sin(-55 ) = 0.82 0.57-55 1 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. For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ). 35

Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems: 36

What are the coordinates of point C? In this example, we know the angle. Using a calculator, we find that cos 44.72 and sin 44.69, so the coordinates of C are approximately (0.72, 0.69). 1 Note that 0.72 2 + 0.69 2 1! 37

The Tangent Function Recall SOH-CAH-TOA sin θ = opp hyp cos θ = adj hyp tan θ = opp adj opposite side hypotenuse θ adjacent side It is also true that tan θ = sin θ cos θ. Why? opp hyp adj hyp opp hyp hyp adj opp adj = = = tan θ 38

Angles in the Unit Circle 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.) 39

Example: Given a terminal point cscθ. Note the "hidden" Pythagorean Triple, 8, 15, 17)., find θ, tanθ and To find θ, use sin -1 or cos -1 : sin -1 ( ) = θ θ 28.1 tanθ = sinθ/cosθ tanθ = cscθ = 1/sinθ cscθ = 40

Example: Find the x-value of point A, θ and the tan θ. For every point on the circle, θ A 22.3-22.3 5 (, - 13 ) Since x is in quadrant III, x = - 12 5 sin -1 (- 13 ) -22.3, BUT θ is in quadrant III, so θ = 180 + 22.3 = 202.3 (notice how 202.3 and -22.3 have the same sine) sin θ tan θ = cos θ = = 5 12 13 41

Example: Given the terminal point of ( -5 / 13, -12 / 13 ). Find sin x, cos x, and tan x. Answer 42

14 What is tan θ? 3 (- 5, ) A θ B C Answer D 43

15 What is sin θ? 3 (- 5, ) A θ B Answer C D 44

16 What is θ (give your answer to the nearest degree)? 3 (- 5, ) θ Answer 45

17 Given the terminal point, find tan x. Answer 46

18 Knowing tan x = Find sin x if the terminal point is in the 2 nd quadrant Answer 47

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. Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two 30-60-90 triangles) 48

Special Right Triangles (see Triangle Trig Review unit for more detail on this topic) 49

Special Triangles and the Unit Circle (-, ) (, ) 1 1-45 45 Multiples of 45 angles have sin and cos of ±, depending on the quadrant. 50

30 o 45 o 60 o 60 o 45 o 30 o 30 o 30 o 45 o 60 o 60 o 45 o 51

Drag the degree and radian angle measures to the angles of the circle: π 5π π 3π 7π 3π 0 π 2π 4 4 2 2 4 4 0 45 90 135 180 225 270 315 360 52

Fill in the coordinates of x and y for each point on the unit circle: (, ) (, ) 3π 4 π 2 π 4 (, ) (, ) π 2π 0 (, ) 0 1 (, ) 5π 4 3π 2 7π 4 (, ) -1 (, ) 53

Special Triangles and the Unit Circle (, ) 1 (, ) 1 30 60 Angles that are multiples of 30 have sin and cos of ± and ±. 54

Drag the degree and radian angle measures to the angles of the circle: 5π π π π 3π 0 2π 4π π 2π 7π 11π 5π 6 2 6 3 3 2 3 6 6 3 0 30 60 90 120 150 180 210 240 270 300 330 360 55

Drag in the coordinates of x and y for each point on the unit circle: (, ) (, ) (, ) π 5π 6 2π 3 π 2 (, ) π 3 (, ) π 6 2π 0 (, ) (, ) (, ) 7π 6 4π 3 (, ) 3π 2 5π 3 (, ) 11π 6 (, ) (, ) 0 1-1 56

Special Angles in Degrees 57

Radian Values of Special Angles 58

Exact Values of Special Angles 59

Put it all together... 60

Exact values of special angles Complete the table below: Degrees 0 30 45 60 90 Radians sin θ cos θ tan θ 61

1 63

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. 64

Example: If tan =, and sin < 0, find sin, cos and the value of. 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) 65

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. sin = = opp -3 adj hyp cos = = Use any inverse trig function to find the angle. tan-1( ) 36.7. Because the angle is in QIII, we need to add 180 + 36.7 = 216.7, so 217. 66

19 A B C D E F G H I J Answer 67

20 A B C D E F G H I J Answer 68

21 A B C D E F G H I J Answer 69

22 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 Answer 70

23 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 Answer 71

24 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 Answer 72

Graphing Trig Functions Return to Table of Contents 73

If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions: (Once the webpage opens, click on Download) 74

Graphing the Sine Function, y = sin x Graphby 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. (Remember, is just a bit more than 3.) Since the values are based on a circle, values will repeat. 75

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. 76

Graphing the Cosine Curve 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. 77

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. 78

Compare the graphs: y = sin x y = cos x How are they similar and how are they different? 79

Characteristics of y = sin x and y = cos x 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. 80

Predict, Explore, Confirm 1. 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. Teacher Notes 3. Do your conclusions match your predictions? 81

y = a sin x or y = a cos x 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. Teacher Notes 82

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? y = sin x y = 2sin x Answer y = sin x Name the amplitude of each graph. 83

As shown in the graph below, the graph of y = -3cos x is 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 The domain of each function is the set of real numbers and the range is {x -3 x 3}. 84

Sketch each graph on the interval from 0 to 2 : y = 4cos x y = -.25 sin x 85

25 What is the amplitude of y = 3cos x? Answer 86

26 What is the amplitude of y = 0.25cos x? Answer 87

27 What is the amplitude of y = -sin x? Answer 88

28 What is the range of the function y = 2sin x? A All real numbers B -2 < x < 2 C 0 x 2 D -2 x 2 Answer 89

29 What is the domain of y = -3cos x? A All real numbers B -3 < x < 3 C 0 x 3 D -3 x 3 Answer 90

30 Which graph represents the function y = -2sin x? A B C D Answer 91

31 What is the amplitude of the graph below? 92

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. 94

y = sin bx or y = cos bx Consider the graphs of y = cos x and y = cos 2x. y = cos x one cycle y = cos 2x Notice that the graph of y = cos 2x completes one cycle twice as fast, or in units. 95

y = cos x completes 1 cycle in 2π. So the period is 2π. y = cos 2x completes 2 cycles in 2π or 1 cycle in π. The period is π. y = cos 0.5x completes a cycle in 4π. The period is 4π. 96

The period for y = cos bx or y = sin bx is P = 2 b 2 1 y = cos x b = 1 P = = 2 y = cos 2x b = 2 P = 2 = 2 2 0.5 y = cos 0.5x b = 0.5 P = = 4 97

32 What is the period of A B C Answer D 98

33 What is the period of A B C D Answer 99

34 What is the period of A B C Answer D 100

Sketch the graph of each function from x = 0 to x = 2. Hint y = 2cos 3x y = cos x y = sin 2x y = -2cos 2x 101

35 What is the period of the graph below? A B 2 C 3 D 2 2 Answer 102

36 What is the period of the graph shown? A B C D 2 2 3 3 Answer 103

37 What is the equation of this function? A B C D y = sin 3x y = cos 3x y = 3cos x y = 3sin x Answer 104

Predict, Explore, Confirm 1. Using your prior knowledge of transforming functions, predict what happens to the following functions: Teacher Notes 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? 105

Translating Sine and Cosine Functions 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? Answer 106

Horizontal or phase shift y = cos x Drag each equation to the matching graph y = cos (x + ) 2 Vertical shift y = sin x y = sin x + 2 k 107

38 What is the phase shift for the following function. Use the appropriate sign to indicate direction. Answer 108

39 What is the phase shift for the following function use the appropriate sign to indicate direction. Answer 109

Consider the graphs of 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. 110

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. 111

40 What is the phase shift for the following function? Use the appropriate sign to indicate direction. Answer 112

41 What is the phase shift for the following function? Use the appropriate sign to indicate direction. Answer 113

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. 114

42 What is the vertical shift in Answer 115

43 What is the vertical shift in Answer 116

44 What is the vertical shift in Answer 117

Putting it all together: Find the amplitude, period, phase shift and vertical shift of the following: Amplitude: -2 = 2 Period: Phase Shift: Vertical Shift: -5 (down 5) 118

Graphing a Sine or Cosine Function: 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. 119

Example: Step 1: Amplitude: -1 = 1 Period: Phase Shift: Vertical Shift: 2 (up 2) 120

Step 2: Draw the midline y = 2 Step 3: Find the 5 key points Note: for x, adding the cycle, 3 by 4. comes from dividing For y, adding and subtracting 1 comes from the amplitude. 121

Step 4: Graph 122

You try: Answer 123

45 What is the amplitude of Answer 124

46 What is the period of A B 2 C D 125

47 What is the phase shift of Answer 126

48 What is the vertical shift of Answer 127

49 What is the amplitude of this cosine graph? Answer 128

50 What is the period of this cosine graph? (use 3.14 for pi) Answer 129

51 What is the vertical shift of this cosine graph? Answer 130

52 Which of the following of the following is an equation for the graph? A B C Answer D 131

The equation y = 4.2cos (π/6(x - 1)) + 13.7 can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, 1-12. Sketch the graph of this equation. What is the average temperature in June? Answer 132

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. 133

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. 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. 134

Example: Sketch the graph of y = tan (x + ) + 2 Asymptotes will be at 0,, 2, etc. The midline will be at y = 2. 135

53 Which graph represents y = -tan x? A B Answer C D 136

Trigonometric Identities Return to Table of Contents 137

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. 138

Drag each equation into the correct box: 3x + 4 = 3x + 4 3x + 4 = 9 5x - 7y = -(7y - 5x) 2x 5 =x 3 sin θ + cos θ = 1 tan θ cot θ =1 2(x-1) = 2x - 2 (x + 3) 2 = x 2 + 9 sin 4x = 4sin x Identities Conditional Equations 139

Reciprocal Identities Basic Trig Identities csc θ = 1 sin θ sin θ = 1 csc θ sec θ = 1 cos θ cos θ = 1 sec θ cot θ = 1 tan θ tan θ = 1 cot θ Tangent Identity tan θ = sin θ cos θ Cotangent Identity cot θ = cos θ sin θ 140

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. 141

Algebraic example Trig example (x - y)(x + y) = x 2 - y 2 (1 - cos θ)(1 + cos θ) = 1 - cos 2 θ 142

Pythagorean Identities Recall the unit circle, x 2 + y 2 = 1. (-1,0) (0,1) 1 (0,-1) (cos θ,sin θ) (1,0) For any point (x, y) on the circle, its coordinates are (cos θ, sin θ). Therefore, (cos θ) 2 + (sin θ) 2 = 1 2, which is usually written as cos 2 θ + sin 2 θ = 1 143

Pythagorean Identities How do we transform the first identity, which is derived from the unit circle, to the other two? 144

Alternative Forms of Identities Since we know that 3 + 5 = 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. 145

More Alternative Forms Another fact family tells that since follows that 4 5 = 20. 20 5 = 4, it 1 Since sec θ = cos θ, then sec θ cos θ = 1 (multiply both sides of the first equation by cos θ). 146

Simplifying Trig Expressions Example 1: Simplify csc θ cos θ tan θ. Rewrite each trig ratio in terms of cos and sin: 1 sin θ sin θ cos θ = 1 cos θ (When multiplying fractions, it is often easier to reduce or cancel before you multiply.) Example 2: Simplify csc 2 θ(1 - cos 2 θ). 1 sin 2 θ (sin 2 θ) = 1 147

Verifying an Identity Transform one side of the identity to be the same as the other side Example 1: Verify sin θ cot θ = cos θ sin θ cos θ sin θ = cos θ Example 2: Verify cos θ csc θ tan θ = 1 1 sin θ sin θ cos θ cos θ = 1 148

Simplify: Answer 149

Simplify: Answer 150

Simplify: Answer 151

Simplify: Answer 152

Simplify: Answer 153

Verify: Answer 154

Verify: Answer 155

Verify: Answer 156

Verify: Answer 157

Verify: Answer 158

54 Which equation is NOT an identity? 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 - 1 159

55 The following expression can be simplified to which choice? A B C D Answer 160

56 The following expression can be simplified to which choice? A B C Answer D 161

57 The following expression can be simplified to which choice? A B C Answer D 162