Algebra 2 Honors: Trigonometry Semester 2, Unit 6: Activity 31

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1 Resources: SpringBoard- Algebra Online Resources: Algebra Springboard Text Unit 6 Vocabulary: Arc length Unit circle Radian Standard position Initial side Terminal side Coterminal angles Reference angle Trigonometric function Periodic function Period Amplitude Midline Phase shift Algebra Honors: Trigonometry Semester, Unit 6: Activity 31 Unit Overview In this unit, students build on their knowledge of trigonometry from geometry and extend it to radian measure and the unit circle. Students will apply trigonometric functions to understanding real-world periodic phenomena. Student Focus Main Ideas for success in lessons 31-1 & 31- Introduce students to radian measurement Use a real-world problem to develop understanding of radian measure and how it differs from degree measure Example: Lesson 31-1: Vocabulary: The arc length is the length of a portion of the circumference of a circle. The arc length is determined by the radius of the circle and by the angle measure that defines the corresponding arc, or portion, of the circumference. When you find the arc length generated by a radius on a circle with radius 1, it is called a unit circle. On a unit circle, the constant of proportionality is the measure of the angle of rotation written in radians, which equals the length of the corresponding arc on the unit circle. Page 1 of 36

2 Example A: A toy train travels 30 around a circular track with a radius of 8 feet. What is the constant of proportionality that can be used to find the distance along the track that the train travels? a) b) c) d) Convert radians to degrees Convert degrees to radians multiply radians by the ratio multiply degree by the ratio Lesson 31-: Example A: A Ferris wheel makes one complete rotation every 4 minutes. How far, to the nearest tenth, will a rider who is seated 40 feet from the center travel in 10 minutes? a) 15.7 feet b) 00 feet c) 51.3 feet d) 68.3 feet Example B: What is 150 in radians? Page of 36

3 Resources: SpringBoard- Algebra Online Resources: Algebra Springboard Text Unit 6 Vocabulary: Arc length Unit circle Radian Standard position Initial side Terminal side Coterminal angles Reference angle Trigonometric function Periodic function Period Amplitude Midline Phase shift Algebra Honors: Trigonometry Semester, Unit 6: Activity 3 Unit Overview In this unit, students build on their knowledge of trigonometry from geometry and extend it to radian measure and the unit circle. Students will apply trigonometric functions to understanding real-world periodic phenomena. Student Focus Main Ideas for success in lessons 3-1 & 3- Calculate trigonometric ratios for acute angles using the ratios of the sides of a right triangle Use reference angles and the unit circle to find trigonometric ratios of any angle Example: Lesson 3-1: Vocabulary: An angle is in standard position when the vertex is placed at the origin and the initial side is on the positive x-axis. The other ray that forms the angle is the terminal side. Draw an angle in standard position with a measure of 10. Since 10 is 30 more than 90, the terminal side is 30 counterclockwise from the positive y-axis. Page 3 of 36

4 Draw an angle in standard position with a measure of -00. Since -00 is negative, the terminal side of 00 clockwise from the positive x-axis. Draw and angle in standard position with a measure of radians. Since is greater than radians, the terminal side makes one full rotation, plus an additional radians. Find one positive and one negative angle that are coterminal with each given angle. If θ is an angle in standard position, its reference angle α is the acute angle formed by the terminal side of θ and the x-axis. The graphs show the reference angle α for four different angles that have their terminal sides in different quadrants. Page 4 of 36

5 The relationship between θ and α is shown for each quadrant when 0 < θ < 360 or 0 < θ < π. Quadrant I Quadrant II Quadrant III Quadrant IV QUESTION: Find the reference angle for. ANSWER: The terminal side of θ lies in Quadrant III, therefore. so QUESTION: Find the reference angle for. When an angle is not between 0 and 360 ( ), find a coterminal angle that is within that range. Then use the coterminal angle to find the reference angle. ANSWER: The terminal side of θ lies in Quadrant II, therefore so. QUESTION: Find the reference angle for. Since 435 is greater than 360, subtract. Now determine the reference angle for 75. ANSWER: Since 75 is in Quadrant I, the reference angle is 75. Page 5 of 36

6 QUESTION: Find the reference angle for radians. Since is greater than, subtract ANSWER: The terminal side of this angle is in Quadrant III so, so. QUESTION: Find the sine and cosine of 90 QUESTION: Find the sine and cosine of 180. Page 6 of 36

7 Lesson 3-: QUESTION: What are the sine and cosine of θ when? The sine and cosine are the lengths of the legs of a If θ is not in the first quadrant, use a reference angle. ANSWER: length of shorter leg length of longer leg triangle. QUESTION: What are sinθ and cosθ when radians? Page 7 of 36

8 Page 8 of 36 Unit Circle:

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10 Resources: SpringBoard- Algebra Online Resources: Algebra Springboard Text Unit 6 Vocabulary: Arc length Unit circle Radian Standard position Initial side Terminal side Coterminal angles Reference angle Trigonometric function Periodic function Period Amplitude Midline Phase shift Algebra Honors: Trigonometry Semester, Unit 6: Activity 33 Unit Overview In this unit, students build on their knowledge of trigonometry from geometry and extend it to radian measure and the unit circle. Students will apply trigonometric functions to understanding real-world periodic phenomena. Student Focus Main Ideas for success in lessons 33-1 & 33-: Use Pythagorean Theorem to prove the Pythagorean Identity, Use the Pythagorean Identity to find given the value of one of those functions and the quadrant of θ. Combine the Pythagorean Identity with the reciprocal identities to prove related Pythagorean identities. Example: Lesson 33-1: Page 10 of 36

11 EXAMPLE: Lesson 33-: Page 11 of 36

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13 Resources: SpringBoard- Algebra Online Resources: Algebra Springboard Text Unit 6 Vocabulary: Arc length Unit circle Radian Standard position Initial side Terminal side Coterminal angles Reference angle Trigonometric function Periodic function Period Amplitude Midline Phase shift Algebra Honors: Trigonometry Semester, Unit 6: Activities 34 & 35 Unit Overview In this unit, students build on their knowledge of trigonometry from geometry and extend it to radian measure and the unit circle. Students will apply trigonometric functions to understanding real-world periodic phenomena. Student Focus Main Ideas for success in lessons 34-1, 34-, 34-3, 34-4, & 34-5: Analyze, graph, and write equations for parent trigonometric functions and their transformations, including phase shifts Identify period, midline, amplitude, and asymptotes Main Idea for success in lesson 35-1: Using knowledge of trigonometric functions and their graphs, model realworld periodic phenomena using functions of the form or Example: Lesson 34-1: Example 1: Period: 3 Amplitude: 4 Midline: y = Page 13 of 36

14 Lesson 34-: Graph of Sine Function Transformations of Sine Function Page 14 of 36

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17 Lesson 34-3: Graph of Cosine Function Page 17 of 36

18 Page 18 of 36 Transformations of Cosine Functions

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21 Lesson 34-4: Graph of Tangent Function Page 1 of 36

22 Page of 36 Transformations of Tangent Function

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25 Page 5 of 36 Lesson 34-5:

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27 Summary of Transformations y = a sin b(x y = a cos b(x y = a tan b(x h) + k h) + k h) + k a The coefficient changes the amplitude of the sine and cosine functions. When a > 1, the amplitude increases and the graph is stretched vertically. When 0 < a < 1, the amplitude decreases and the graph is compressed vertically. When a < 0, the graph is reflected across the x-axis. b The coefficient changes the period. When b > 1, the period decreases and the graph is compressed horizontally. When 0 < b < 1, the period increases and the graph is stretched horizontally. The period of sin bx and cos bx is πb. The period of tan bx is πb. h The constant shifts the graph horizontally. When h > 0, the graph shifts to the right. When h < 0, the graph shifts to the left. k The constant shifts the graph vertically. When k > 0, the graph shifts up. When k < 0, the graph shifts down. Page 7 of 36

28 Lesson 35-1: Example 1: Hector s skateboard wheels have a diameter of 80 millimeters. When he starts riding, a small chip in one wheel is visible at the top of the wheel. As he rides, the wheels make 6 revolutions per second. What is the function that gives the height h in millimeters of the chip in the wheel as a function of time t in seconds? Example : The tide at Lookout Point is modeled by height h in feet as a function of time t in hours since low tide., giving Which describes the heights of low and high tide and the time in between them? Page 8 of 36

29 Name class date Lesson Going around a circle you travel the length of an arc formed by a 45 angle. If the distance you travel is 0 m, what is the radius of the circle? Algebra Unit 6 Practice 5. Reason quantitatively. A passenger sits 0 m from the center of a Ferris wheel and travels a quarter of a turn. What is an accurate estimate of the distance traveled in meters?. Make sense of problems. Two people are sitting in different locations on a merry-go-round. The first person sits 4 m from the center of the ride and the second person sits 6 m from the center of the ride. How much farther has the second person traveled than the first person when they complete a turn? Lesson Convert the following angle measures from radians to degrees. a. p 7 b. 3 p 8 c d Write the constant of proportionality for each of the following angles in a unit circle. Express your answers in terms of p. a. 3 b. 18 c. 54 d Make sense of problems. The measure of linear velocity is meters per second and the measure of angular velocity is radians per second. Angular velocity describes how fast an object in circular motion travels in terms of the angle measure of the arc (in radians) that the object travels in a second. If you are on a merry-go-round sitting 4 m from p the center and you travel radians per second, 9 what is your linear velocity? In other words, what is the length of the arc that you travel every second? 4. Which of the following constants of proportionality in a unit circle is paired with its corresponding angle? 8. Model with mathematics. Come up with a general formula to convert angular velocity v to linear velocity v in terms of the radius r. A. 1 p 4p, 0 B., C. p 14p, 4 D., College Board. All rights reserved. 1 SpringBoard Algebra, Unit 6 Practice Page 9 of 36

30 Name class date 9. It takes a Ferris wheel 1 minutes to complete a turn. How many radians does it turn every minute? A B. 0.5 C. p 7 D. 9 p Make sense of problems. Imagine looking at a Ferris wheel from the side so that we can divide it into four quadrants. A passenger starts at the very bottom of the Ferris wheel, and the Ferris wheel makes a complete turn counterclockwise every 10 minutes. What quadrant will the passenger be in at 14 minutes? Explain. 10. A Frisbee rotates 40 times every two seconds. How many radians does it rotate every second? A. 1 p 7 B. 0p C. 40p D. 10p Lesson Which of the following angle pairs are coterminal angles? A. p p 54, 5 4 B. 5 p 4, p 4 C. 9 p 4, 3p D. 9 p 4 4, 7p 4 1. Which of the following angle pairs are coterminal with 33? A. 13, 147 B. 71 p 30, 49p 30 C. 393, 37 D. 71p p 60, Reason quantitatively. Are angles measuring 68p 88 and radians coterminal? Explain your 45 answer. 15. What are the reference angles for the following angles? a. 78 b. 68 c. 3 p 8 Lesson Find the sine of each of the following angles. a. 585 b. 750 c Find the cosine of each of the following angles. a. 585 b. 750 c What is the tangent of 480? A. 1 B. 0 C. 3 D College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 30 of 36

31 Name class date 19. Reason abstractly. The sum of the internal angles of a triangle is 180. If a right triangle has one angle that is 45, what can be deduced about the length of the two shorter sides of the triangle? 0. Express regularity in repeated reasoning. Complete the following table Given that sin u 5 3 8, and 0, u, p, the values of cos u and tan u? A. cos u B. cos u C. cos u D. cos u and tan u and tan u and tan u and tan u what are sine cosine Reason abstractly. Why can the value of the tangent function exceed 1 when the values of the sine and cosine functions cannot? tangent Lesson 33-1 p 1. Given that 0, u,, find the value of cos u for each of the given values of sin u. a. sin u Make sense of problems. A right triangle has a hypotenuse of length 8 m. If one of the angles of the triangle is 30, what are the lengths of the other two sides of the triangle? b. sin u c. sin u p. Given that 0, u,, each of the given values of cos u. find the value of sin u for Lesson Simplify the following expression. csc u sin u 1 cot u a. cos u 5 3 b. cos u c. cos u College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 31 of 36

32 Name class date 7. Simplify the following expression. sin u tan u cot u 8. Given that sin u 5 3, 0, u, p, values of cos u and sec u? what are the Lesson Is this graph a periodic function? If so, find the amplitude. The maximum value of the function is 7 and the minimum value is 7. y 10 5 A. cos u 5 B. cos u and sec u 5 35 and sec u x C. cos u 5 D. cos u and sec u and sec u Reason abstractly. The cosecant, secant, and tangent functions are the reciprocals of the sine, cosine, and tangent functions respectively. What is always the product of the multiplication of each of these functions by its reciprocal? 3. Is this graph a periodic function? If so, find the amplitude. The maximum value of the function is 39 and the minimum value is y Make sense of problems. Does sin x (cos x) (sec x) 5 sin x 1? Explain your answer x College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 3 of 36

33 Name class date 33. Reason qualitatively. Is the function represented by the graph below a periodic function? Why or why not? y 10 Lesson Use appropriate tools strategically. Complete the table below and use it to construct a graph of the function y 5 3 sin x. x sin x y 5 3 sin x x 0 p 3 p p Attend to precision. Estimate the period and the amplitude for the function graphed below. y 10 p 4p 3 3p 5p 3 p 37. What is the amplitude, midline and period of the 5 function y sin x? x 38. What is the sine function that has an amplitude of 5 6, a midline of y 5 0, and a period of p? What is the maximum value of the function f(x) 5 4 cos (x)? A. ` B. 0 C. 4 D Which of the following sine functions describes a graph that has a period of 8p, an amplitude of, and a midline of y 5 1? A. y 5 sin (4x) 1 1 B. y 5 sin (x) 1 1 x C. y 5 sin x D. y 5 sin College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 33 of 36

34 Name class date 40. Attend to precision. Find the value of y when p x x 5 in the function y 5 4 sin Use appropriate tools strategically. Complete the table below and use it to construct a graph of the function y 5 cos x. 3 x cos x y 5 3 cos x 0 Lesson Attend to precision. Find the value of y when x x 5 4p in the function y 5 4 cos Write the equation for the cosine function that has an amplitude of, a period of p, and a midline of 3 y 5 0. p 3 p p 3 p 4p 3 3p 5p 3 p 43. What are the amplitude, midline, and period of the function y cos 1 5 x? Lesson What is the range of the function y 5 3 tan (x) 1 1? 44. Which are the maximum and minimum values for the function f (x) 5 3 cos (x) 1 1? 47. Reason qualitatively. Given that tan x 5 sin x cos x and that the domains for both sin x and cos x are all real numbers, why are there gaps in the graph for y 5 tan x? A. maximum 5 3, minimum 5 3 B. maximum 5 5 3, minimum C. maximum 5 5 3, minimum Attend to precision. Find the value of y when x x 5 4p in the function y 5 5 tan D. maximum 5 3, minimum College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 34 of 36

35 Name class date 49. Name the period, zeros, and asymptotes of the function y tan 3 7 x. 50. Which of the following is not a zero for the function y 5 tan 3 x? A. x 5 p 3 3p C. x 5 Lesson 34-5 B. x 5 3 p D. x List the amplitude and period and describe the horizontal and vertical shifts relative to the parent function of y 5 3 sin 4 p x Which are the features of the graph of y 5 5 tan 3 3p x 1 4? A. amplitude: 5, period: p 3, horizontal shift: 3p right, vertical shift: 4 up p 3p B. period:, horizontal shift: right, 3 vertical shift: 4 up C. amplitude: 5, period: p 3, horizontal shift: 3p left, vertical shift: 4 up p 3p D. period:, horizontal shift: right, 3 vertical shift: 4 down 54. Reason abstractly. The function y 5 cot x is the reciprocal of the function y 5 tan x. Describe how the graph of y 5 cot x compares to the graph of y 5 tan x. 5. List the amplitude and period and describe the horizontal and vertical shifts relative to the parent function of y cos p x Attend to precision. Find the value of y when x x 5 4p in the function y 5 4 cos 4 1 sin x x 1 tan College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 35 of 36

36 Name class date Lesson Write a trigonometric function that describes the height as a function of time of a car on a Ferris wheel that makes a complete rotation every minutes. The radius of the Ferris wheel is 5 m and at its highest point the car is 60 m high. 59. A car engine is running at 6000 revolutions (turns) per minute. Which of the following could describe the position of a point on a gear attached directly to the engine as a function of time t in seconds? A. h(t) sin p t Attend to precision. What is the height of the car in Item 56 after 7 minutes and 1 seconds? B. h(t) sin (00pt) C. h(t) sin 60p t 6000 D. h(t) sin p t Make sense of problems. What is the height of the car in Item 56 at its lowest point? 60. The distance of an object from the ground in meters can be expressed as a function of time in seconds by the function h(t) 5 7 cos p t What is 7 the maximum height of this object, and when does the object reach that height for the first time? College Board. All rights reserved. SpringBoard Algebra, Unit 6 Practice Page 36 of 36