# Semester 2, Unit 4: Activity 21

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1 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 21 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 21-1: Define the reciprocal and quotient identities. Use and transform the Pythagorean Identity. Example Lesson 21-1: Page 1 of 21

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3 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 22 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 22-1 and 22-2: Use the unit circle to write equivalent trigonometric equations. Write cofunction identities for Sine and Cosine. Solve trigonometric equations using identities and by graphing. Example Lesson 22-1: Math Tip: To convert degrees to radians, multiply the degree measure by Page 3 of 21

4 Example Lesson 22-2: Page 4 of 21

5 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 23 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 23-1, 23-2, and 23-3: Use Sum and Difference Identities. Use Half Angle Identity. Derive the identities and use them to find the exact values of trigonometric functions. Use trigonometric identities to solve equations. Example Lesson 23-1: Page 5 of 21

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7 Example Lesson 23-2: Sum and Difference Identities Page 7 of 21

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9 Example Lesson 23-3: Page 9 of 21

10 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 24 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 24-2: Write equations for the Law of Cosines using a standard angle. Apply Law of Cosines in a real world problem. Example Lesson 24-2: Law of Cosines. The Law of Cosines is useful in many applications involving non-right triangles, also known as oblique triangles. Page 10 of 21

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12 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 25 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 25-1 and 25-2: Discover mathematical relations to derive the Law of Sines. Find unknown sides and angles of an oblique triangle by using the Law of Sines. Example Lesson 25-1: Like the Law of Cosines, the Law of Sines relates the sides and angles in an oblique triangle, and these can be used to find unknown sides or angles given at least three known measures that are not all angle measures. Law of Sines Page 12 of 21

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16 NAME CLASS DATE LEssON Make use of structure. Express each trigonometric function as a reciprocal to write an identity. u a. sec 2 Precalculus Unit 4 Practice 5. Make sense of problems. Suppose you know that cos u a. Using the Pythagorean identity, what can you conclude about sin u? ( ) b. tan x 12p b. How does your answer to part a change if you know that u lies in Quadrant III? 2. Make use of structure. Express each trigonometric function as a quotient to write an identity. 2p a. cot 3 b. tan (x 2 308) 3. Make use of structure. Express each trigonometric function using the Pythagorean Theorem to write an identity. 2 p a. tan c. What is the approximate value of u, in degrees, given that u lies in Quadrant III? LEssON Which expression is equivalent to tan u csc u? A. B. cosu sin 2 u 1 sec u C. csc u D. sec u b. csc 2 (x ) 2 sec t 1 7. Reason abstractly. Simplify sec 22. t 4. Consider the function f( x) 5sin 2 x1 cos 2 x. What is 23 f 181? A B C D College Board. All rights reserved. 1 SpringBoard Precalculus, Unit 4 Practice Page 16 of 21

17 NAME CLASS DATE Attend to precision. For Items 8 and 9, verify each identity. sin x 8. 5cosx tanx 13. Reason abstractly. Use the unit circle to find the value of each of the following. P(c, d ) 1 9. (1 2 cos 2 a)(1 1 cot 2 a) Simone attempted to verify the identity sin 2 u csc u sec u 5 tan u by entering sin 2 u csc u sec u 2 tan u as function Y 1 in the graphing calculator. Is her method correct? Explain. a. cos (2p 2 u) b. sin ( u) LEssON 22-1 For Items 11 and 12, complete each statement. p 11. If cos x 5 0.8, then sin x A. 0.8 B C D p 14. Given that sin ى 0.34, 9 name three additional angles whose sines are approximately equal to Verify the identity cos ( u) tan u 5 2sin u. Be sure to justify each step of your argument with a valid reason. 12. Make use of structure. If tan x 5 1.2, then tan (p 2 x) College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 17 of 21

18 NAME CLASS DATE LEssON Attend to precision. Solve each equation over the given interval without a calculator. 1 a. 2 x2 x5 2 cot cos 0, p, 32 p b. 4 tan 2 a 5 sin 2 a, 0, p 2 LEssON 23-1 The sound of a single musical note can be represented using the function y 5 a sin (2p ft), where a is the amplitude (volume) of the sound measured in decibels (db), f is the frequency (pitch) of the note measured in hertz (Hz), and t is time. 17. What is the solution of cos 2 x over the interval [08, 3608)? A. 58 B C. infinite number of solutions D. no solution 21. Model with mathematics. The frequency of the musical note middle G is about 208 Hz. Write the equation for the sine wave that represents middle G played at a volume of 55 db. 22. Write the equation for the sine wave for the note one octave below middle G played at a volume of 80 db. 18. Use appropriate tools strategically. Solve the equation over the given interval. You may use a calculator. 3 tan 2 u , [08, 3608) 19. Evan solved the equation cos x 5 21 sin x on the interval p 2, p. His work is shown below. Explain his error and find the correct solution. 23. Make use of structure. Write the equation for the sine wave that represents the sound when both notes in Items 21 and 22 are played at the same time. 24. Use a calculator to graph the functions = ٣, y 1 ى ٢ x ١ 2 cos 2 52, and y 1 1 y 2 on the interval [22p, 2p], and then make a sketch in the space below. Step 1: cos x 5 21 sin x Original equation cos x Step 2: 521 sin x Divide both sides by cos x. Step 3: tan x 5 21 Definition of tangent 3p Step 4: x 5 4 Solve for x. 20. Explain how the answer to tan 2 x 51, [08, 3608) could not be an infinite number of solutions. 25. Express the function y 1 1 y 2 in the form y 5 sin (x 1 c) College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 18 of 21

19 NAME CLASS DATE LEssON Make sense of problems. In Item 25, you found the function y 1 1 y 2 in the form y 5 sin (x 1 c). Use the sum identity for sine to algebraically verify this identity. LEssON 23-3 Make use of structure. Verify each identity. 31. sin (x 2 p) 5 2sin x 27. Make use of structure. Find the exact value. a. sin 15 p 6 b. tan (1058) ( ) cos u 1 a tanu tan a 5 cosu cos a Given sin a5 and tan b52 with angle 5 10 a terminating in Quadrant I and angle b terminating in Quadrant II, what is the exact value of sin (a 1 b)? A. 7 5 B. C D Attend to precision. Solve each equation on the interval [08, 3608]. 33. sin 2u 2 cos u cos 4 u 2 sin 4 u 5 cos 2u Given sin A5 and tan B52 with /A 5 10 terminating in Quadrant I and /B terminating in Quadrant II, find the exact value of tan (A2B) Given cot θ5 6 with 0 < u ڤ p 2 sin 2u, and tan 2u., find sin u, cos u, 35. Determine the number of solutions on the interval [0, 2p ) for cos (4u) A. 2 B. 4 C. 6 D College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 19 of 21

20 NAME CLASS DATE LEssON 24-1 For Items 36 40, refer to Item 17a in Lesson Make sense of problems. Explain why the speed of the blade is at 0 ft/sec at the value(s) you found. Model with mathematics. For Items 37 40, find the distance from the center of the wheel to the stirrer blade for each angle From Ghirardelli Square in San Francisco, you can see the Golden Gate Bridge and Alcatraz Island. The angle between the sight lines to these landmarks is approximately 808. The approximate distance from Ghirardelli Square to the Golden Gate Bridge is 3.2 miles and to Alcatraz is 1.4 miles. A surveyor precisely measured the angle between the sight lines to be By how many miles does the approximate distance change from the Golden Gate Bridge to Alcatraz? At which value of u does the speed of the blade reach a maximum? 43. The sides of an isosceles triangle have lengths of 7.2, 7.2, and What are the angles of the triangle? A. 778 B C D A triangle has side lengths of 8, 10, and 12. What are the angles of the triangle? LEssON Model with mathematics. A new courtyard at Ghirardelli Square in San Francisco will be triangular, as shown in the diagram below. Retaining Wall 45 yd 30 yd 1058 Suppose the angle opposite of the retaining wall needs to be decreased by 78. If the lengths of the courtyard are to remain the same, find the length of the retaining wall and the new angle measurement. 45. A triangle has two side lengths of 1 2 and 2 and an 3 angle measure of 328 between them. Which of the following is closest to the length of the remaining side? 1 A B C D College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 20 of 21

21 NAME CLASS DATE LEssON 25-1 For Items 46 48, refer to Items 1 and 2 in Lesson Model with mathematics. Suppose the plane traveled on its diverted path instead of turning course towards Honolulu after 1.5 hours. a. How many more miles would the plane have traveled after 2 hours? LEssON Determine how many triangles are possible with the given information: 428, b 5 60, a A. no triangle B. one triangle C. two triangles D. three triangles b. How far would the plane have been from Honolulu after 2 hours? 47. Which equation could be used to find the bearing of the plane if it had traveled for 2 hours and needed to turn at that point towards Honolulu? sin x sin20 A sin x sin20 B sin x sin20 C sin x sin20 D Construct viable arguments. Determine how many triangles are possible with the given information. Draw a sketch and show any calculations you used. a , b , a b. 60, b55 3, a Make use of structure. Solve the two-solution ambiguous case situation given b 5 20, a 5 25, B Attend to precision. Find the new bearing of the plane when it turns towards Honolulu after 2 hours. 49. In triangle DEF, angle D is 428, angle E is 788, and DE is Find angle F, EF, and DF. For Items 54 55, solve each triangle using the Law of Sines. 54. A 5 538, B 5 658, c In triangle STU, ST is 25, TU is 30, and SU is Find all three angles. 55. B 5 298, C 5 158, b College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 21 of 21

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