Trigonometry  Chapter 4 Test Solutions  Fall ‘06

1.      For arclength  extending counterclockwise along the unit circle from (1,0)

a.       The reference number for t is   b.      The coordinates of the terminal point P(x,y) are then :  c.       The diagram at right illustrates this point’s position on a plot of the unit circle.
2.      Suppose  highlight that interval on the perimeter of the unit circle a.       If , then, as shown in the diagram,  This is equivalent to  . b.      If , then, as shown in the diagram,  This is equivalent to

 

3.      In the diagram, P is the terminal point for  and Q is the reflection of P across the x-axis.  a.       Since the directed arc from A to Q,  has the same length: .  But this means that the arc  so that  whence the corresponding chords are also equal and  is equilateral. b.      The coordinates of Q are (x,y). c.

d.      Since (x,y) is on the unit circle,  and we can to substitute for y to get .  Expand, collect and get zero on one side: , which is factorable:  

e.       If x =  1/2 then y = .

4. 
   
    implies
   
   
   
   
5. In quadrant II   
   
6. The amplitude, period and phase shift of
   
     is 2, the period is
   
    and the phase shift is -1/16.  Here’s a graph showing two periods.
7. The sinusoid shown below has y values ranging from -1 to 5 so the line of equilibrium is y = 2 and the amplitude is 3.  There’s a peak at x = -1 and the next is at x = 7, so the period is 8.  The wave is at equilibrium at x = 5 so the phase shift is -3.  Thus the sinusoid can be expressed either as a sine or a cosine:
   
   
   

8.      Consider the function .

a.       Vertical asymptotes can be found by solving  where k is any integer:
, whence adjacent asymptotes can be found by plugging in, say, k = 0 and k = 1:  

b.      The distance between asymptotes is one wavelength, so we add a quarter wavelength to the first asymptote three times to get .  Thus the points to plot are  

c.       The graph at right shows how the function passes through the three points and approaches the vertical asymptotes.

9.      Suppose sin t = 3/5 and t is in the first quadrant.  Find the following:
a.               b.        c.