periodic functions, algebra homework help

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Name Date Periodic Functions and Trigonometry Unit Review Answer Key ALGEBRA 2 B Directions: Check your answers to the Periodic Functions and Trigonometry Unit Review. 1. Is the function periodic? If so, what is its period? Answer: Yes, its period is 4. Each cycle occurs in intervals of 4. 2. Find the amplitude of the periodic function. Answer: The amplitude is 2. This is half the distance between the maximum values (3) and the minimum values (−1) . 3. Sketch the angle 225° in standard position. Answer: © © 2014 Connections Education LLC. All rights reserved. 4. What are the sine and cosine of −150° ? Answer: sin(−150°) = − 1 2 cos(−150°) = − 3 2 5. What is the radian measure of an angle of 132° ? Answer: 132° • π 180° = 11π 15 ( 2π radians) and 6. What are the exact values of sin Answer: 3 cos( 2π radians) ? 3 3 1 ; − 2 2 7. What is length a to the nearest tenth? Answer: a = rθ a = 10( 3π ) 4 a = 23.6 m 8. The equation of the graph is of the form y = a sin x . What is the amplitude of the sine curve? What is the value of a? © © 2014 Connections Education LLC. All rights reserved. 2 Answer: 1.5; a = 1.5 9. What is the graph of one cycle of a sine curve with amplitude 5, period π , and a > 0 ? What is an equation for the sine curve in the form y = a sin bθ ? Answer: y = 5sin 2θ 10. Sketch the graph of y = 1 cos 3θ in the interval from 0 to 2π . What are the 2 domain, range, period, and amplitude of the function? Answer: domain: all real numbers; range: − amplitude: 1 1 2π < y < ; period: ; 2 2 3 1 2 11. A water level varies 50 in. between low tide at 8:30 A.M. and high tide at 3:00 P.M. What cosine function models the variation in inches above and © © 2014 Connections Education LLC. All rights reserved. 3 below the average water level as a function of the number of hours since 8:30 A.M.? Answer: Amplitude is ½(50) = 25. Since the tide is at −25 inches at time zero, a = −25 . The cycle is halfway complete after six and a half hours, so the full cycle is 13 hours. 13 = b= 2π b 2π 13 So the function that models the water level is f (t ) = −25cos ( 12. What is the value of tan − π 3 2π t 13 ) ? Do not use a calculator. Answer: − 3 1 2 13. Sketch two cycles of the graph of y = tan θ Answer: ( 3π ) ? Do not use a calculator. 14. What is the exact value of csc Answer: 4 2 15. Use your calculator in degree mode to find the decimal value, rounded to the nearest thousandth, of sec 59° . Answer: 1.942 © © 2014 Connections Education LLC. All rights reserved. 4 16. What is the graph of y = sec x in the interval from 0 to 2π ? Answer: 17. Verify the identity sin 2 θ cot 2 θ = 1 − sin 2 θ . sin 2 θ cot 2 θ = sin 2 θ cos 2 θ sin 2 θ = cos 2 θ = 1 − sin 2 θ 18.What is a simplified trigonometric expression for 1 − cos 2 θ ? tan θ Answer: 1 − cos 2 θ tan θ = sin 2 θ tan θ = sin 2 θ cos θ sin θ = sin θ cos θ 19. What is cos −1 − ( 2 ) in degrees? 2 Answer: 135° 20. What value(s) in the interval θ (−π ≤ θ ≤ π ) satisfy the equation 6sin θ − 2 = sin θ ? Approximate your answer(s). Answer: 0.412 and 2.73 © © 2014 Connections Education LLC. All rights reserved. 5
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Explanation & Answer

HiPlease check the attached file for details, let me know if you have any questons,thank you.

1.
Yes
Period = 4, cycles repeats after 4.
2. Amplitude is 2, which is ½ (max-min).
3.
225 = 180 +45, so the end of the angle is 45 passing – x axis.

225

4.
sin(-150) =- sin(150)= -sin(180�...


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