Math 12 – Precalculus – fall ’04 – Chapter 5 Test Solutions.

1.      Suppose a terminal point determined by t is the point . 

a.       Verify that the point lies on the unit circle.   b.      What are the coordinates of the terminal point for ?  This is the point a half circumference farther along the circle: . c.       What are the coordinates of the terminal point for ?  This is the point a quarter circumference farther along the circle: .

 

2.      Suppose a terminal point determined by t is P(x,y) on the unit circle, where . 

a.       What quadrants could P be in?
SOLN:  P could be in either quadrant II or quadrant IV.

b.      What are the absolute values of the coordinates of x and y ?
SOLN:  Plugging into the Pythagorean identity,  so that  and thus .

c.       Find absolute values for csc(t), and cot(t). 
SOLN:

3.      Suppose a terminal point P(x,y) in QIV on the unit circle has y-coordinate . Find

a.         SOLN: 

b.        SOLN:

4.      Find the reference number for each and plot its position on the unit circle together with exact values for its x and y coordinates.

a.        so that  and b.       where

 

5.      Suppose that .  Estimate the corresponding intervals for the values of  cos(t) and sin(t) and highlight these on the diagram: SOLN:  As shown on the diagram, it appears that if  then  and .

6.      Find the amplitude, period and phase shift of the  and sketch a graph showing at least one wave form.  Be careful scale and label axes in your graph.
SOLN:  Amplitude = 2. Period = 1 and phase shift is 1/12. 

7.      Find the period and at least two asymptotes and graph the function  .
SOLN:  The period is  and the asymptotes are where

8.      Find sinusoidal formula which fits the graph shown below:
SOLN:  Amplitude = 1.  Phase angle = .  Period =