1 CCA Integrated Math 3 Module 4 Honors & 3.3, 3.5 Trigonometric Functions Ready, Set, Go! Adapted from The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2014 Mathematics Vision Project MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.
2 Name Modeling with Geometry 3.3H Ready, Set, Go! Ready Topic: Finding missing measurements in triangles Use the given figure to answer the questions. Round your answers to hundredths place. Questions 1-8 all refer to the same triangle below. Given: 1. Find and 2. Find 3. Find BC 4. Find BA 5. Find CD 6. Find AD 7. Find BD 8. Find the area of
3 Set Topic: Triangle relationships in the special right triangles Fill in all of the missing measures in the triangles. Express answers in simplest radical form. 9. 10. 11. 12. 13. 14. Use an appropriate triangle from above to fill in the function values below. 15. 16. 17. 18. In question 17, does it matter if you used the triangle in question 10, 11, or 12? Explain.
4 Topic: Finding areas of triangles Find the area of each triangle. 19. 20. 21. 22. 23.
5 Topic: Using right triangle trigonometry to solve triangles 24. While traveling across a flat stretch of desert, Joey and Holly make note of a mountain peak in the distance that seems to be directly in front of them. They estimate the angle of elevation to the peak as 5. After traveling 6 miles toward the mountain, the angle of elevation is 25. Approximate the height of the mountain in miles and in feet. 5,280 ft = 1 mile (While figuring, use at least 4 decimal places.) 25. The Star Point Ranger Station and the Twin Pines Ranger Station are 30 miles apart along a straight scenic road. Each station gets word of a cabin fire in a remote area known as Ben s Hideout. A straight path from Star Point to the fire makes an angle of 34 with the road, while a straight path from Twin pines makes an angle of 14 with the road. Find the distance, d, of the fire from the road.
6 Set Topic: The laws of sine and cosine Law of Sines: If ABC is a triangle with sides a, b, and c, then Law of Cosines: If ABC is a triangle with sides a, b, and c, then or it can be written as: Use the Law of Sines to solve each triangle. Round angles to the nearest degree and side lengths to the nearest hundredth. 26. 27. 28. 29. 30. What information do you need in order to use the Law of Sines?
7 Use the Law of Cosines to solve each triangle. Round angles to the nearest degree and side lengths to the nearest hundredth. 31. 32. 33. 34. 35. What information do you need in order to use the Law of Cosines to solve a triangle?
8 Name Modeling with Geometry 3.5H Ready, Set, Go! Ready Topic: Trigonometry ratios of the special triangles Fill in the missing angle. Do not use a calculator. 1. 2. 3. 4. 5. 6. 7. 8. 9. Topic: Verifying trigonometric identities Verify each trigonometric identity. Show all of your steps. 10. 11.
9 12. 13. Set Topic: Area formulas for triangles Area of an Oblique Triangle: The area of any triangle is one-half the product of the lengths of two sides times the sine of their included angle. Find the area of the triangle having the indicated sides and angle. Round answers to the nearest hundredth. 14. 15. 16. 17.
Find the area of these triangles. You will need to find additional measurements first. Round answers to the nearest hundredth. 18. 19. 10 Perhaps you used the Law of Cosines to establish the following formula for the area of a triangle. The formula was known as early as circa 100 B.C. and is attributed to the Greek mathematician, Heron. Heron s Area Formula: Given any triangle with sides of lengths a, b, and c, the area of the triangle is: ( )( )( ) where s is half of the perimeter of the triangle, or ( ). Find the area of the triangle having the indicated sides. Round to the nearest hundredth. 20. 21. 22. 23.
11 Go Topic: Distinguishing between the Law of Sines and the Law of Cosines Indicate whether you would use the Law of Sines or the Law of Cosines to solve the triangles. Then solve each triangle. Round angles to the nearest degree and side lengths to the nearest hundredth. 24. 25. Topic: Solving oblique triangles 26. Find all missing dimensions and angle measures of the given triangle. 27. Use the diagram to answer the question. In applying the Law of Cosines to the figure, a student writes ( )( ). Did the student start the problem correctly? Explain your answer. Solve for by continuing the student s work or starting correctly yourself.
Name: Trigonometric Functions 4.1H Ready, Set, Go! Ready Topic: Coterminal angles State a negative angle of rotation that is coterminal with the given angle of rotation. Coterminal angles share the same terminal side of an angle of rotation. Sketch and label both angles. 12 Example: is the given angle of rotation. The angle of rotation is indicated by the solid arc. The dotted angle of rotation is the coterminal angle with a rotation of 240. 1. Given Coterminal Angle 2. Given Coterminal Angle 3. Given Coterminal Angle
13 Set Topic: Sine and cosine of radian measures 4a. Write the exact coordinates of the points on the circle below. Be mindful of the numbers that are negative. 4b. Find the arc length, s, from the point ( ) to each point around the circle. Record your answers as decimal approximations to the nearest thousandth.
14 Set Topic: Fundamental trigonometric identities The cofunction identities state: ( ) and ( ) Complete the statements, using the Cofunction identities. 5. 6. 7. 8. 9. 10. Use the sum identities for sine and cosine to find the exact value of each. 11. 12. 13. Let and lies in quadrant II. a. Use the Pythagorean identity,, to find the value of. b. Use the Quotient identity,, the given information, and your answer in part a to calculate the value of. 14. Let and lies in quadrant IV. a. Find using the Pythagorean identity b. Find using the quotient identity. c. Find ( ) using a cofunction identity.
15 Use your calculator to find the following values. 15. Why are both of your answers to questions 15 positive? 16. Why are both of your answers to questions 16 negative? 17. In which quadrants are sine and cosine both negative? 18. Name an angle of rotation where sine is equal to. 19. Name an angle of rotation where cosine is equal to. Go Topic: Inverse trigonometric functions Use your calculator to find the value of where. Round your answers to the nearest degree. 20. 21. 22. 23.
Name: Trigonometric Functions 4.2H Set, Go! Set Topic: Using trigonometric ratios to solve problems Perhaps you have seen The London Eye in the background of a recent James Bond movie or on a television show. When it opened in March of 2000, it was the tallest Ferris wheel in the world. The passenger capsule at the very top is 135 meters above the ground. The diameter is 120 meters. 1. How high off the ground is the center of the Ferris wheel? 16 2. How far from the ground is the very bottom passenger capsule? 3. Assume there are 32 passenger capsules, evenly spaced around the circumference. Find the height from the ground of each of the even numbered passenger capsules shown in the figure. Use the figure at the right to help you think about the problem. Topic: Finding trigonometric ratios Let be an acute angle of a right triangle. Find the values of the other five trigonometric functions. 4. 5.
17 Use trigonometric identities to show that the two sides are equivalent. Show all of your steps. 6. 7. ( )( ) 8. ( ) 9. 10. ( )( ) 11.
18 Go Topic: Trigonometric values of the special angles Find two solutions of the equation. Give both answers in degrees ( ( ). Do not use a calculator. ) and radians 12. 13 degrees: radians: degrees: radians: 14 15 degrees: radians: degrees: radians: 16. 17. degrees: radian: degrees: radians:
Name: Trigonometric Functions 4.3H Ready, Set, Go! Ready Topic: Using graphs to evaluate function expressions 19 Use the graph of ( ) and ( ) below to find the indicated values. 1. ( ) ( ) 2. ( ) ( ) 3. ( ) ( ) 4. ( ) ( ) ( ) ( ) 5. ( ) ( ) Topic: Finding the length of an arc using proportions Use the given degree measure of the central angle to set up a proportion to find the length of minor arc AB or the length of the semicircle. Leave your answers in terms of. Recall that ( ) where s is the arc length. 6. 7. 8. The circumference of circle A is 400 meters. The circumference of circle B is 800 meters. What is the relationship between the radius of circle A and the radius of circle B? Justify your answer
20 Set Topic: Solving trigonometric equations Find all solutions of the equations in the interval ). 9. 10. 11. 12. Find the general solutions of the equations below. 13. 14.
21 15. ( ) ( ) Go Topic: Finding missing angles in triangles Find the measure of each acute angle for the triangle at the right. Round your answers to the nearest degree. 16. 17. 18. 19.
Name: Trigonometric Functions 4.4H Ready, Set, Go! Ready Topic: Describing intervals of graphs Write the intervals where the graph is positive and the intervals where the graph is negative. 22 1. Positive: Negative: 2. Positive: Negative: 3. (The scale on the x-axis is in increments.) Positive: Negative:
23 4. (The scale on the x-axis is in 45 increments.) Positive: Negative: Write the piecewise equations for the given graphs. 5. 6. Set 7. Sketch the graph of a Ferris wheel that has a radius of 1 unit, with center at a height of 0, and makes one complete rotation in 360 seconds.
24 Topic: Graphing trigonometric functions Graph the following functions (the graph of is given to assist you). 8. 9. Go 10. A submarine at the surface of the ocean makes an emergency dive, its path making an angle of with the surface. a. If it goes 300 meters along its downward path, how deep will it be? b. What horizontal distance is it from its starting poing? c. How many meters must it go along its downward path to reach a depth of 1000 meters?
Name: Trigonometric Functions 4.5H Ready, Set, Go! Ready Topic: Comparing radius and arc length The wheels on the wagons that the pioneers used to cross the plains were smaller in the front than in the back. The front wheel had 12 spokes. The top of the front wheel measured 44 inches from the ground. The rear wheel had 16 spokes. The top of the rear wheel measured 59 inches from the ground. (For these problems disregard the hub at the center of the wheel. Assume the spokes meet in the center at a point.) Find the following values. Express answers in terms of. 1. Find the area and the circumference of each wheel. 25 2. Determine the central angle between the spokes on each wheel. 3. Find the distance on the circumference between two consecutive spokes for each wheel. 4. The wagons could cover a distance of 15 miles per day. How many more times would the front wheels turn than the back wheels on an average day? 5. A wheel rotates r times per minute. Describe how to find the number of degrees the wheel rotates in t seconds.
26 Set Graph the following functions (the graph of or is given to assist you). 6. 7. 8. 9. 10. ( ) 11. ( )
27 Go Topic: Solving problems using right triangle trigonometry Make a sketch of the following problems and then solve. 12. A kite is flying, attached at the end of a string that is 1500 feet long. The string makes an angle of 43 with the ground. How far above the ground is the kite? Round your answer to the nearest foot. 13. A ladder leans against a building. The top of the ladder reaches a point on the building that is 12 feet above the ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the measure of the angle that the ladder makes with the level ground. What is the angle the ladder makes with the building? 14. The shadow of a flagpole is 40.6 meters long when the angle of elevation of the sun is 34.6. Find the height of the flagpole. Round your answer to the nearest tenth of a meter.
Name Trigonometric Functions 4.6H Set, Go! Set Topic: Values of sine in the coordinate plane Use the given point on the circle to find the value of sine. Recall that and. 1. 2. 28 3. In each graph above, the angle of rotation is indicated by an arc and. Describe the angles of rotation that make the y-values of the points be positive and the angles of rotation that make the y-values be negative. 4. In the graph at the right, the radius of the circle is 1. The intersections of the circle and the axes are labeled. Based on your observations from the task, what is the value of sine might be for: 90? 180? 270? 360?
29 Topic: Values of cosine in the coordinate plane Use the given point on the circle to find the value of cosine. Recall that and. 5. 6. 7. In each graph, the angle of rotation is indicated by an arc and. Describe the angles of rotation that make the x-values of the points be positive and the angles of rotation that make the x-values be negative. 8. In the graph at the right, the radius of the circle is 1. The intersections of the circle and the axes are labeled. Based on your observation in #10, what is the value of cosine might be for: 90? 180? 270? 360?
30 Describe the relationships between the graphs of ( ) (solid) and equations. 9. 10. ( ) (dotted). Then write their 11. This graph could be interpreted as a shift or a reflection. Write the equations both ways. 12.
31 Sketch the graph of the function. Include 2 full periods. Label the scale of your horizontal axis. 13. ( ) 14. ( ) 15. ( ) 16.
32 Go Topic: Measures in special triangles is a right triangle. Angle C is the right angle. Use the given information to find the missing sides and the missing angles. 17. 18. 19. 20. Find AD in the figures below: 21. 22.
Name: Trigonometric Functions 4.7H Ready, Set, Go! Ready Topic: Functions and their inverses Indicate which of the following functions have an inverse that is a function. If the function has an inverse, sketch it. Remember, the inverse will reflect across the line. Finally, label each one as even, odd, or neither. Recall that an even function is symmetric with respect to the y-axis, while an odd function is symmetric with respect to the origin. 1. 2. 33 3. 4.
34 Set Topic: Graphs of the trigonometric functions State the period, amplitude, vertical shift, and phase shift of the function shown in the graph. Then write the equation using the given trigonometric parent function. 5. 6. Period: Amplitude: Period: Amplitude: Vertical shift: Phase shift: Vertical shift: Phase shift: Equation: Equation: 7. 8. Period: Amplitude: Period: Amplitude: Vertical shift: Phase shift: Vertical shift: Phase shift: Equation: Equation:
35 9. 10. The cofunction identity states that ( ) and ( ). How does this identity relate to the graph in #9? Explain where you would see this identity in a right triangle. Period: Vertical shift: Amplitude: Phase shift: Equation: Go Name two values (if possible) for the angle of rotation, ratio., that have the given trigonometric 11. 12. 13. 14. 15. 16.
36 17. For which angles of rotation does? Explain why. Topic: Finding angle measures given trigonometric ratios in a right triangle. Given the trigonometric values from a right triangle, find the indicated angle measures. Round angle measures to the nearest degree. Hint: draw a right triangle and label the lengths of the sides using the trigonometric ratio. 18. 19. 20. Go Topic: Trigonometric ratios in a right triangle Find the other two trigonometric ratios based on the one that is given. Hint: draw and label a right triangle using the given trigonometric ratio. 21. 22. 23. 24. 25. 26.
Name Trigonometric Functions 4.8H Ready, Set, Go! Ready Topic: Using the definition of tangent Use what you know about the definition of tangent in a right triangle and your work with the new definitions of sine and cosine to find the exact value of tangent for each of the right triangles below. 1. 2. 37 3. 4. 5. In each graph above, the angle of rotation is indicated by an arc and. Describe the angles of rotation from 0 to that make tangent be positive and the angles of rotation that make tangent be negative.
38 Set Topic: Mathematical modeling using sine and cosine functions Many real-life situations such as sound waves, weather patterns, and electrical currents can be modeled by sine and cosine functions. The table below shows the depth of water (in feet) at the end of a wharf as it varies with the tides at various times during the morning. t (time) midnight 2 A.M. 4 A.M. 6 A.M. 8 A.M. 10 A.M. noon d (depth) 8.16 12.16 14.08 12.16 8.16 5.76 7.26 We can use a trigonometric function to model the data. Suppose you choose cosine: ( ), where y is the depth at any time. The amplitude will be the distance from the average of the highest and lowest values. This will be the average depth, d. 6. Sketch the line that shows the average depth, d. 7. Find the amplitude: ( ) 8. Find the period:. Since a normal period for sine is, the new period for our model will be so. Use the p you calculated, divide and turn it into a decimal to find the value of b. 9. High tide occurred 4 hours after midnight. The formula for the displacement is. Use b and solve for c. 10. Now that you have your values for a, b, c, and d, you can put them into your equation: ( ) 11. Use your model to calculate the depth at 9 A.M. and 3 P.M. 12. A boat needs at least 10 feet of water to dock at the wharf. During what interval of time in the afternoon can it safely dock?
39 Topic: Transformations of trigonometric graphs Match each trigonometric representation on the left with an equivalent representation on the right. Then check your answers with a graphing utility. Record your answer in the space provided to the left of the question number. 13. ( ) A. 14. ( ) B. 15. C. 16. D. 17. ( ( )) E. ( ) 18. ( ) F. ( )
40 Go Topic: Using the calculator to find angles of rotation Use your calculator and what you know about where sine and cosine are positive and negative in the unit circle, to find the two angles that are solutions to each equation. Make sure is in the interval. Round your answers to 4 decimal places. (Your calculator should be set in radians.) You will notice that your calculator will sometimes give you a negative angle. That is because the calculator is programmed to restrict the angle of rotation so that the inverse of the function is also a function. Since the requested answers have been restricted to positive rotations, if the calculator gives you a negative angle of rotation, you will need to figure out the positive coterminal angle for the angle that your calculator has given you. 19. 20. 21. Note: When you ask your calculator for the angle, you are undoing the trigonometric function. Finding the angle is finding the inverse trigonometric function. When you see ( ), you are being asked to find the angle that makes true. The answer would be the same as the answer your calculator gave you in #19. Another notation that represents the inverse sine function is ( ). Topic: Graphing sine and cosine functions Graph the following functions (the graphs of and are given to assist you). 22. ( ) Hint: there will be a horizontal translation in this graph 23.
Name Trigonometric Functions 4.9H Ready, Set, Go! Ready Topic: Rigid and non-rigid transformations of functions 41 The equation of a parent function is given. Write a new equation with the given transformations. Then sketch the new function on the same graph as the parent function. If the function has asymptotes, sketch them in. 1. Vertical Shift: down 8 Horizontal Shift: left 6 Dilation: Equation: Domain: Range: 2. Vertical Shift: up 3 Horizontal Shift: left 4 Dilation: Equation: Domain: Range:
42 3. Vertical Shift: none Horizontal Shift: left 5 Dilation: Equation: Domain: Range: 4. Vertical Shift: up 1 Horizontal Shift: left Dilation (amplitude): 3 Equation: Domain: Range: Set Topic: Features of the graphs of the trigonometric functions 5. is a right triangle with. Use the information in the figure to label the length of the sides and measure of the angles. 6. is an equilateral triangle. and is an altitude Use the information in the figure to label the length of the sides, RA, and the exact length of. Label the measure of and.
43 7. Use the information from the figures in questions 6 and 7 to fill in the table. Function 8. Name the angles of rotation, in radians, where sine equals 0 in the domain ]. 9. Name the angles of rotation, in radians, where cosine equals 0 in the domain ]. 10. Name the angles of rotation, in radians, where tangent equals 0 in the domain ]. 11. Name the angles of rotation, in radians, where tangent is undefined in the domain ]. Topic: Graphing tangent functions. Graph each tangent function. Be sure to indicate the location of the asymptotes. 12. ( ) 13. ( )
44 Go Topic: Trigonometric facts Answer the questions below. Be sure you can justify your thinking. 14. Given triangle ABC with being the right angle, what is? 15. Identify the quadrants in which is positive. 16. Identify the quadrants in which is negative. 17. Identify the quadrants in which is positive. 18. Explain why it is impossible for. 19. Name the angles of rotation, in radians, for when. 20. Which trigonometric function has the same value when the angle of rotation is positive or negative? 21. Explain why ( ) 22. Write the Pythagorean identity and then prove it. 23. Explain why, in the unit circle,. 24. Which function represents the slope of the hypotenuse in a right triangle? 25. Name the trigonometric function(s) that are odd functions.
45 Topic: Graphing sine and cosine Sketch the graph of each function. Include two full periods. Label the scale of your horizontal axis. 26. ( ) 27. ( ) 28. ( ) 29. ( )
46 Name: Trigonometric Functions 4.9H Ready, Set, Go! Ready Topic: Function combinations The functions ( ) ( ) ( ) are given in the graphs below. Graph the indicated combination on the same axes. 1. ( ) ( ) 2. ( ) 3. ( ) ( ) Set Topic: Graphing reciprocal trigonometric functions Graph each function. Remember, it may be helpful to sketch sine, cosine, or tangent first. Use the space below the function to construct a table of values to use for your graph. 4. ( ) x ( ) ( )
47 5. ( ) x ( ) ( ) 6. ( ) x ( ) 2+3 ( ) 7. ( ) x ( ) ( )
Write the equation of the given graphs. Check the accuracy of your equations using a graphing utility. 8. Use 9. Use 48 Go Topic: Finding all six trigonometric ratios Use the given information to find the exact value of all remaining trigonometric ratios of. Hint: It may be helpful to draw a diagram. 10. 11. 12. 13.
Name: Trigonometric Functions 4.13H Ready, Set, Go! Ready Topic: Graphing trigonometric functions Match the equation with the correct graph. A. B. ( ) C. D. ( ) E. F. 1. 2. 49 3. 4. 5. 6.
50 Set Topic: Inverse trigonometric functions Use the given information to find the missing angle ( place. ). Round answers to the thousandths 7. ; is in Quadrant IV 8. ; 9. ; 10. ; 11. ; 12. ; 13. ; 14. 15. Explain why #14 needed only 1 clue to determine a unique value for, and the others required at least 2 clues. Graph each function. 16. 17. ( )
51 Topic: Using the calculator to find angles of rotation Use your calculator and what you know about where sine and cosine are positive and negative in the unit circle, to find the two angles that are solutions to each equation. Make sure is in the interval. Round your answers to 4 decimal places. (Your calculator should be set in radians.) You will notice that your calculator will sometimes give you a negative angle. That is because the calculator is programmed to restrict the angle of rotation so that the inverse of the function is also a function. Since the requested answers have been restricted to positive rotations, if the calculator gives you a negative angle of rotation, you will need to figure out the positive coterminal angle for the angle that your calculator has given you. 18. 19. 20. Topic: Composite trigonometric functions Recall that a composite function places one function such as ( ), inside the other, ( ), by replacing the x in ( ) with the entire function ( ). In general, the notation is ( ( )). It is possible to do composition of the trigonometric functions. The answer to ( ) is an angle of. The composition of ( ( ) ) is simply asking What is value of?" The answer is. Sine was just undoing what problems 27 32. was doing. Not all composite trigonometric functions are inverses such as Answer the following. For questions 28-32, it may help to draw a diagram. 21. ( ) 22. ( ) 23. ( ) 24. ( ) 25. ( ) 26. ( ) 27. ( ) 28. ( ) 29. ( )
52 For each function, identify the amplitude, period, horizontal shift, and vertical shift. Then sketch a graph. 30. ( ) ( ( )) amplitude: period: horizontal shift: vertical shift: 31. ( ) ( ) amplitude: period: horizontal shift: vertical shift: Go Topic: Verifying trigonometric identities Half of an identity and the graph of this half are given. Use the graph to make a conjecture as to what the right side of the identity should be. Then prove your conjecture. 32. ( ) 33.
53 Shifty Functions A Solidify Understanding Task Part I Given the function ( ) and information about ( ), Complete the table for ( ). Graph ( ) on the same set of axes as the graph of ( ). Write a brief conclusion statement for the changes you notice. ( ) 5 3 1 0 1 2 1. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement:
54 2. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) 0 Conclusion statement: 3. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement:
55 4. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement: 5. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement:
56 6. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement: 7. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement:
57 8. If ( ) ( ), use the domain ] to complete the table and graph of ( ). ( ) Conclusion statement: Part II: Communicate your understanding. 9. If ( ) ( ), describe the relationship between ( ) and ( ). 10. If ( ) ( ), describe the relationship between ( ) and ( ). 11. Describe how negatives change the shape of the graph when comparing the graphs of ( ), ( ), and ( ). 12. Describe how multiplication by a constant,, changes the shape of the graph when comparing the graphs of ( ), ( ), and ( ).