Trigonometry, Exam 2 Review, Spring (b) y 4 cos x

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1 Trigonometr, Eam Review, Spring 8 Section.A: Basic Sine and Cosine Graphs. Sketch the graph indicated. Remember to label the aes (with numbers) and to carefull sketch the five points. (a) sin (b) cos Section.B: Sine and Cosine Graphs. Sketch the graph of the function over one complete period. State the amplitude, period, and whether there is an -ais reflection, a shift up/down, or a phase shift. Remember to label our aes carefull. sin Amplitude: Is there an -ais reflection? Yes or No Period: Shift up or down: Phase Shift:. Sketch the graph of the function over one complete period, starting at the phase shift and showing at least one period to the right of it. State the amplitude, period,

2 Page and whether there is an -ais reflection, a shift up/down, or a phase shift. Remember to label our aes carefull. cos Amplitude: Is there an -ais reflection? Yes or No Period: Shift up or down: Phase Shift:. The graph of one complete period of a cosine curve is given. Find the amplitude, period, phase shift, and whether there is a shift up/down. Then, write an equation that represents the curve in the form c acos k( b). Amplitude = Is there an -ais reflection? Period: Shift up or down: Phase shift: Equation: Refer to the graph to answer the questions. The equation representing the

Page curve can be written in the form c acos k( b) (a) What is the period of this graph? (b) What is the amplitude of this graph? (c) What is the value of c, the vertical shift?

3 Page curve can be written in the form c acos k( b) (a) What is the period of this graph? (b) What is the amplitude of this graph? (c) What is the value of c, the vertical shift?.. A cork floating in a lake is bobbing up and down. The height of the cork above 8.cos t, where t is the bottom of the lake is modeled b the equation the time in minutes and is the height, measured in meters. (a) What is the amplitude for this graph? What does it represent in this situation? (b) What is the period of this graph? What does it represent in this situation? (c) What are the maimum and the minimum heights of the cork? (d) How man periods does the bobbing cork complete each minute? 7. A mill has a water wheel that is ft in diameter and rotates counterclockwise at a rate of revolution ever seconds. The bottom of the wheel is ft above the surface of the water. Let h be the height (in feet) of the point P above the water at time t seconds. Assume the point P is at the o clock position on the wheel when it starts rotating, as indicated in the figure. (a) Determine the height of point P above the water level at the following times. Then, plot these points on the aes given. t = time (sec.). 7. h = height (ft) P (b) Assuming this situation can be modeled b a sine or a cosine equation, find an appropriate equation to represent the height h of point P at time t seconds, b finding each of the following. Remember, equations of this form can be written as: c asink or c acosk.

4 Amplitude = -ais reflection? Yes or No What is the value of a? Period = What is the value of k? Vertical shift: units Up or Down (circle one) What is the value of c? Equation: (c) What is the height of point P after the wheel has been turning for seconds? Round to one decimal. Section.: Tangent Graphs 8. Find the period and phase shift of each of the following. (a) tan (b) tan (c) tan8 9. Determine whether there is a vertical stretch or compression, the period, and whether there is an -ais reflection for the graph of tan. Then, graph the function over at least two periods. Label the aes and asmptotes appropriatel. Page Vertical stretch or compression factor: -ais reflection? es or no Period = Sections./.: Inverses. Find the eact value (in radians) of each epression, if it is defined. (a) sin (b) tan ( ) (c) cos

5 . Find the angle θ, in degrees, indicated in the triangle. Round to decimal place.. Write as an algebraic epression in, free of trigonometric or inverse trigonometric functions:. m θ tancos,.8 m Page. A -ft ladder is leaning against a building. It touches the building 9 feet up from the ground. What is the measure of the angle (in degrees) the ladder forms with the ground? Round to decimal.. A basketball plaer is standing feet from the basketball goal. The rim is feet above the ground level. The basketball plaer s ees are.8 feet above the floor. What is the angle of elevation (in degrees) from the plaer s ees to the rim? Round to decimal. For each of the following, find the eact value, if it is defined. Remember, ou must be able to show our work or give a reason on each problem. 7. coscos. sin sin 7. tan tan 8. cos cos 9. sintan. costan. tan cos. cos sin

6 Page ANSWERS:. (a) (b) Amp. = ; es; period = 8; up ; none Amp. = ; es; period = π; phase shift π Amp. = ; es; period = π; no shift up/down; phase shift = π; equation: cos. (a) (b) (c). Amplitude =.; The cork rises and falls b. meters. (b) Period = /; It takes / of a minute for the cork to complete its up/down ccle. (c) Maimum = 8. meters; minimum = 7.8 meters (d) The bobbing cork completes periods (bobbing up/down) each minute. 7. (a) t. 7. h (b) Amp= ; no -ais reflection; a = ; period = ; k = π/; vertical shift units up; c = ; equation h sin t (c) h() sin. ft

7 Page 7 8. (a) period = ; phase shift = (b) period = ; phase shift = (c) period = ; phase shift = 8 9. Vertical stretch b a factor of ; Period = ; Yes, there is an -ais reflection. (See graph at right. Points shown are at (, ), (, ), and (, ); asmptotes are labeled.) / = - = - = =. (a) (b) does not eist. 7. (c)

TRANSFORMING TRIG FUNCTIONS

Chapter 7 TRANSFORMING TRIG FUNCTIONS 7.. 7..4 Students appl their knowledge of transforming parent graphs to the trigonometric functions. The will generate general equations for the famil of sine, cosine