Slide 52 / 162 Drag the degree and radian angle measures to the angles of the circle: # # 5# 7# 3# 3# # 2# 0 4 2 4 4 4 2 0 ∘ 45 ∘ 90 ∘ 135 ∘ 180 ∘ 225 ∘ 270 ∘ 315 ∘ 360 ∘
Slide 53 / 162 Fill in the coordinates of x and y for each point on the unit circle: ( , ) # ( , ) 2 3# ( , ) # 4 4 ( , ) 0 ( , ) 0 # 2# 1 5# 4 -1 7# 3# ( , ) 2 4 ( , ) ( , )
Slide 54 / 162 Special Triangles and the Unit Circle ( , ) ( , ) 1 1 30 ∘ 60 ∘ Angles that are multiples of 30 have sin and cos of ± and ± .
Slide 55 / 162 Drag the degree and radian angle measures to the angles of the circle: # # 2 # 3# 4# # 2# 5# 7# 11# 5# 2# 0 2 3 3 3 6 6 6 6 3 0 ∘ 30 ∘ 60 ∘ 90 ∘ 120 ∘ 150 ∘ 180 ∘ 210 ∘ 240 ∘ 270 ∘ 300 ∘ 330 ∘ 360 ∘
Slide 56 / 162 Drag in the coordinates of x and y for each point on the unit circle: ( , ) # ( , ) ( , ) 2 # 2# 3 3 ( , ) ( , ) # 5# 6 6 0 # 2# ( , ) ( , ) 0 11# 7# ( , ) 1 6 6 3# ( , ) 4# 5# 2 3 -1 3 ( , ) ( , ) ( , )
Slide 57 / 162 Special Angles in Degrees
Slide 58 / 162 Radian Values of Special Angles
Slide 59 / 162 Exact Values of Special Angles
Slide 60 / 162 Put it all together...
Slide 61 / 162 Exact values of special angles Complete the table below: Degrees 0 ∘ 30 ∘ 45 ∘ 60 ∘ 90 ∘ Radians sin θ cos θ tan θ
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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.
Slide 65 / 162 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. -3 · Draw a right triangle in adj Quadrant III. opp · Use the Pythagorean hyp Theorem to find the length of the hypotenuse: (Continued on next slide)
Slide 66 / 162 Once we know the lengths -3 for each side, we can adj calculate the sin, cos and opp the angle. Used the signed hyp numbers to get the correct values. sin = = 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 .
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Slide 70 / 162 22 Which functions are positive in the second quadrant? Choose all that apply. cos x A sin x B tan x C sec x D csc x E cot x F
Slide 71 / 162 23 Which functions are positive in the fourth quadrant? Choose all that apply. cos x A sin x B tan x C sec x D csc x E cot x F
Slide 72 / 162 24 Which functions are positive in the third quadrant? Choose all that apply. cos x A sin x B tan x C sec x D csc x E cot x F
Slide 73 / 162 Graphing Trig Functions Return to Table of Contents
Slide 74 / 162 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)
Slide 75 / 162 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 y or sin x . of , and the corresponding values of (Remember, is just a bit more than 3.) Since the values are based on a circle, values will repeat.
Slide 76 / 162 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 77 / 162 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.
Slide 78 / 162 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 79 / 162 Compare the graphs: y = cos x y = sin x How are they similar and how are they different?
Slide 80 / 162 Characteristics of y = sin x and y = cos x amplitude = 1 range: -1 ≤ y ≤ 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.
Slide 81 / 162 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. 3. Do your conclusions match your predictions?
Slide 82 / 162 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.
Slide 83 / 162 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 = 2sin x y = sin x y = sin x Name the amplitude of each graph.
Slide 84 / 162 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}.
Slide 85 / 162 Sketch each graph on the interval from 0 to 2 : y = -.25 sin x y = 4cos x
Slide 86 / 162 25 What is the amplitude of y = 3cos x ?
Slide 87 / 162 26 What is the amplitude of y = 0.25cos x ?
Slide 88 / 162 ? y = -sin x 27 What is the amplitude of
Slide 89 / 162 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
Slide 90 / 162 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
Slide 91 / 162 30 Which graph represents the function y = -2sin x ? B A C D
Slide 92 / 162 31 What is the amplitude of the graph below?
Slide 93 / 162 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. 3. Do your conclusions match your predictions?
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.
Slide 95 / 162 bx or y = cos y = sin bx Consider the graphs of y = cos x and y = cos 2x . y = cos x y = cos 2x one cycle Notice that the graph of y = cos 2x completes one cycle twice as fast, or in units.
Slide 96 / 162 #. So the period is 2π. y = cos x completes 1 cycle in 2 # or 1 cycle in #. The period is #. y = cos 2x completes 2 cycles in 2 y = cos 0.5x completes a cycle in 4 #. The period is 4 #.
Slide 97 / 162 The period for y = cos bx or y = sin bx is 2 2 P = b b 2 2 P = = 2 y = cos x b = 1 1 2 y = cos 2x b = 2 P = = 2 2 y = cos 0.5x b = 0.5 P = = 4 0.5
Slide 98 / 162 32 What is the period of A B C D
Slide 99 / 162 33 What is the period of A B C D
Slide 100 / 162 34 What is the period of A B C D
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