Domain Test Review Questions - Chapter 4 - Trigonometric Function Properties, Identities and Parametric Functions

Chapter 4 - Trigonometric Function Properties, Identities and Parametric Functions

Chapter Review Questions pp. 166-168

Review Problems:

R0. Journal Updates:

R1. Figure 4-7a shows a unit circle and an angle theta in standard position.
a. Explain why u 2 + v 2 = 1.
b. Explain why u = cos theat and v = sin theta.
c. Explain why cos 2 theta + sin 2 theta = 1.
d. Give a numerical example that confirms the property in part C.
e. Plot on the same screen y1 = cos2 theta and y2 = sin 2 theta. Sketch the graphs. How do the graphs support the Pythagorean property cos 2 theta + sin 2 theta = 1?

R2.
a. Write equations expressing tan x and cot x in terms of sin x and cos x.
b. Write equations expressing tan x and cot x in terms of sec x and csc x.
c. Write three equations in which the product of 2 trig gunction equals 1.
d. make a table of values showin numerically that cos2 x + sin 2 x = 1.
e. Write equations expressing
1. sin 2 x in terms of cos x
2. tan 2 x in terms of sec x
3. csc 2 x in terms of cot x
f. Sketch the graph of the parent function y = cos x. On the same set of axes, skethc the graph of y = sec x using the fact that secant is the reciprocal of cosine.
R3

a. Transform tan A sin A + cos A to sec A. What values of A are exclude from the domain?

b. Transform (cos B _ sin B)2 to 1 + 2 cos B sin B. What values of B are excluded from the domain?
c. Transform 1/1 + sin C + 1/1-sin C to 2 sec 2 C. What values of C are excluded from the domain?
d. Prove the equation
csc D (csc D - sin D) = cot 2 D is an identity. What values of A are excluded from the domain?
e. Prove that the equation (3 cos E = 5 sin E)2 + (5 cos E -3 sin E)2 = 34 is an identity.
f. Showe that the two expressions in part b are equivalent by plotting each on your grapher.
g. Make a table of values to whow that the equation in part e is an identity.

R4. a Find the general solution for theta = arcsin 0.3.
b. Solve 1 + tan 2pi (x + 0.6) = 0 algebraically for the first four positive values of x. Confirm graphically that your solutions are correct.
c. Solve (2 cos theta - 1)(2 sin theta + sq rt of 3) = 0 algebraically in the domain theta as an element [0, 540].Confirm graphically that your solutions are correct.

R5. Plto the graph of this parametric function on your grapher. Sketch the result.
x = -2 + 5 cos t
y = 1 + 3 sin t
b. Use the Pythagorean property for cosine and sine to eliminate the parameter in part a.
c. How can you conclude from the answer to part b that the graph is an ellipse? Where is the center of the ellipse? What are the x and y radii?
d. Figure 4-7B shows a solid cone in perpsective. Write parametric equations for the ellipse that represents the circular base of the cone. Draw the cone on your grapher.

R6.
a. Using parametric mode on your grapher, duplicate the graph of the circular relation y = arccos x shown in Figure 4-7c.

b. Sketch the graph of y = cos -1 x , the principal branch of y = arccos x. Explain the specifications used for selecting this principal branch. What is the range of this inverse cosine function?

c. How is the graph of y = arccos x related to the graph of y = cos x?
d. Find geometrically the exact value (no decimals) of sin (tan-1 2). Check the answer by direct calculations.
e. Write an equation for y = tan (cos-1_x) that does not involve trigonometric functions.
Confirm your answer by plotting it together with the given function on the same screen.
Sketch the result.
f. Prove that cos (cos-1 x) = x.
g. Show on a uv-diagram the range of values of the functions sin-1 and cos-1.
h. Explain why the prefix arc- is appropriate in the names arccos, arcsin, and so on.

Concept Problems

C1. Pendulum Problem:

a.
b.
c.
d.

C3. Square of a Sinusoid Problem:

a.
b.

c.