Domain Test Review Questions- Chapter 3--Applications of Trigonometric and Circular Functions

Applications of Trigonometric and Circular Functions:

R0. Update your journal with what you have learned since the last entry. Include such things as

The one most important thing you have learned as a result of studying this chapter

The graphs of the six trigonometric functions

How the transformations of sinusoidal graphs relate to function transformations in Chapter 1

How the circular and trigonometric functions are related

Why circular functions are usually more appropriate as mathematic models than are trigonometric functions

R1. Sketch the graph of a sinusoid. On the graph, show the difference in meaning between a cycle and a period. Show the amplitude, the phase displacement, and the sinusoidal axis.

b. In y = 3 + 4 cos 5(theta - 10 degrees), what name is given to the quantity 5(theta - 10)?

R2. a. Without using your grapher, show that you understand the effects of the constants in a sinusoidal equation by sketchin the graph of y = 3 + 4 cos 5( theta-10 degrees). Give the amplitude, period, sinusoidal axis location, and the phase displacement.

b. Using the cosine function, find a particular equation for the sinusoid in Figure 3-8a. Find another particular equation using the sine function. Show that the equations are equivalent to each other by plotting them on the same screen.
What do you observe about the two graphs?
Figure 3-8a?

c. A quarter-cycle of a sinusoid is shown in Figure 3-8b. Find a particular equation for it.

d. At what value of theta shown in Fig 3-8b does the graph have a critical point?
e. Find the frequency of the sinusoid shown in Figure 3-8b.

R3. a. Sketch the graph of y = tan theta.
b. Explain why the period of tangent is 180 degrees rather than 360 like sine and cosine.
c. Plot the graph of y = sec theta on your grapher. Explain how you did this.
d. Use the relationship between sine and cosecant to explain why the cosecant function has vertical asymptotes at theta = 0, 180, 360...
e. Explalin why the graph of cosecant has high and low points but no points of inflection.
Explain why the graph of cotangent has points of inflection but no high or low points.
f. For y = 2 + 0.4 cot 1/3 (theta - 40), give the vertical and horizontal dilations and the vertical and horizontal translations. Then plot the graph to confirm that your answers are right.
What is the period of this function? Why is it not meaningful to talk about its amplitude?

R4.
a. How many radians in 30 degrees? 45? 60? Write tne answers exactly in terms of pi.
b. How many degrees in an angle of 2 radians?
c. Find cos 3 and cos of 3 degrees.
d. Find the radian measure of cos -1 0.8 and csc -1 2.
e. How long is the arc of a circle subtended by a central angle of 1 radian if the radius of the circle is 17 units?

R5.
a. Draw a unit circle in a uv-coordinate system. In this coordinate system, draw an x-axis vertically with its origin at the point (u.v) = (1, 0). Show where the points x = 1, 2 and 3 units map onto the units circle as the x-axis is wrapped around it.

b. How long is the arc of a unit circle subtended by a central angle of 60? Of 2.3 radians?
c. Find sin 2 degrees and sin 2.
d. Find the value of the inverse trigonometric function cos-1 0.6.
e. Find the exact values (no decimals) for the circular functions cos pi/6, sec pi/4, and tan pi/2.
f. Sketch the graphs of the parent circular functions y = cos x, and y = sin x.
g. Explain how to find the period of teh circular function y = 3 + 4 sin pi/10(x-s) from the constants in the equation. Sketch the graph. Confirm by your grapher that your sketch is correct.
h. Find a particular equation for the circular function sinusoid for which half a cycle is shown in Figure 3-8c.

R6. Find the generla solution of the inverse circular relation arccos 0.8.
b. Find the first three positive values of the inverse circular relation acrocos 0.8.
c. Find the least value of arccos 0.1 that is greater than 100.
d. For the sinusoid in Fig. 3-8d, fin the four value sof x shown for whih y = 2
Graphically to 1 decimal plce
Numerically by finding the particular equation and plotting the graph
Algebraically, using the particular equation

e. what is the next positive value of x for which y = 2, beyond the last positive value shown in Figure 3-8D?

R7. Porpoising Problem: assume that you are aboard a research submarine doing submerged training exercises in the Pacific Ocean. At time t = 0 you start porpoising (alternately deeper and then shallower). At time t = 4 min you are at your deepest, y = - 1000 min. At time t = 9 min you reach your next shallowest, y = -200 m. Assume that y varies sinusoidally with time.

a. Sketch the graph of y versus t.
b. Find an equation expressing y as a function of t.
c. Your submarine can't communicate with ships on the surface when it is deeper than y = -300 m. At time t = 0 could your submarine communicate? How did you arrive at your answer?
d. Between what two nonnegative times is your submarine first unable to communicate?

Concept Problems:

C1 Pump Jack Problem.

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b.

c.

d.

e.

f.

g.

h.

C3: Angular Velocity Problem:

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c.

d.