Name class date Precalculus Unit 4 Practice Lesson 21-1 5. Make sense of problems. Suppose you know that cos u 5 20.25. 1. Make use of structure. Express each trigonometric function as a reciprocal to write an identity. a. Using the Pythagorean identity, what can you conclude about sin u? u a. sec 2 b. How does your answer to part a change if you know that u lies in Quadrant III? b. tan ( x 1 2p ) c. What is the approximate value of u, in degrees, given 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 Lesson 21-2 6. Which expression is equivalent to tan u csc u? b. tan (x 2 308) 3. Make use of structure. Express each trigonometric function using the Pythagorean Theorem to write an identity. A. cos u sin 2 u B. 1 sec u C.\tcsc u p a. tan 1 1 7 2 D.\tsec u b.\tcsc2 (x 1 41.258) sec 2 t 2 1 7. Reason abstractly. Simplify . sec 2 t 4. Consider the function f ( x )5 sin 2 x 1 cos 2 x . 23 What is f ? 181 A.\t0.99 B.\t1.00 C.\t1.12 D.\t1.25 2015 College Board. All rights reserved. 1 SpringBoard Precalculus, Unit 4 Practice Name class date 13. Reason abstractly. Use the unit circle to find the value of each of the following. Attend to precision. For Items 8 and 9, verify each identity. sin x 8. 5 cos x tan x P (c, d ) 1 \u001f 9.\t(1 2 cos2 a)(1 1 cot2 a) 5 1 10. Simone attempted to verify the identity sin2 u csc u sec u 5 tan u by entering sin2 u csc u sec u 2 tan u as function Y1 in the graphing calculator. Is her method correct? Explain. a. cos (2p 2 u) b. sin (3608 2 u) Lesson 22-1 For Items 11 and 12, complete each statement. p 0.34, name three additional 9 angles whose sines are approximately equal to 0.34. 14. Given that sin p 11. If cos x 5 0.8, then sin 2 x 5____. 2 A. 0.8 B. 20.8 C.\t1.25 15. Verify the identity cos (1808 2 u) tan u 5 2sin u. Be sure to justify each step of your argument with a valid reason. D. 21.25 12. Make use of structure. If tan x 5 1.2, then tan (p 2 x) 5 . 2015 College Board. All rights reserved. 2 SpringBoard Precalculus, Unit 4 Practice Name class date Lesson 22-2 Lesson 23-1 16. Attend to precision. Solve each equation over the given interval without a calculator. 1 3p a. 2 cot x 2 cos x 5 0, p , 2 2 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. p b. 4 tan2 a 5 sin2 a, 0, 2 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. 17. What is the solution of cos2 x 1 5 5 0 over the interval [08, 3608)? A.\t58 B.\t5.99978 22. Write the equation for the sine wave for the note one octave below middle G played at a volume of 80 dB. C. infinite number of solutions D. no 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. 18. Use appropriate tools strategically. Solve the equation over the given interval. You may use a calculator. 3 tan2 u 2 1 5 0, [08, 3608) 24. Use a calculator to graph the functions 1 3 y1 = sin x , y 2 52 cos x , and y1 1 y2 on the 2 2 interval [22p, 2p], and then make a sketch in the 19. Evan solved the equation cos x 5 21 sin x on the p interval , p . His work is shown below. 2 Explain his error and find the correct solution. space below. Step 1: cos x 5 21 sin x Original equation cos x 521 sin x Step 3: tan x 5 21 Step 2: Step 4: x 5 3p 4 Divide both sides by cos x. Definition of tangent Solve for x. 20. Explain how the answer to tan2 x 51, [08, 3608) could not be an infinite number of solutions. 2015 College Board. All rights reserved. 25. Express the function y1 1 y2 in the form y 5 sin (x 1 c). 3 SpringBoard Precalculus, Unit 4 Practice