Math 5  Trigonometry  Chapter 4 Test  fall ’07  Name_________________________

Show your work for credit.  Write all responses on separate paper.  Don’t use a calculator.

 

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

a.       Find the reference number for t.

b.      Find the coordinates of the terminal point P(x,y).

c.       Illustrate this point’s position on a plot of the unit circle.




2.      Consider the point  

a.       Verify that the point lies on the unit circle. 

b.      Use the diagram at right to approximate to the nearest tenth a value of t  so that

c.       Approximate to the nearest tenth a value of t so that  

 




3.      Suppose that  and  point and sin(t) < 0. 
Find  



4.      Write  in terms of , assuming the terminal point for t is in quadrant III.



5.      Find the amplitude, period and phase shift of , construct a table of values and graph one period of the function, clearly showing the position of key points.

6.      Find an equation for the sinusoid whose graph is shown:

7.      Consider the function .

a.       Find the equations for two adjacent vertical asymptotes and sketch them in with dashed lines. 

b.      Find the x-coordinates where y = 0 and where .

c.       Carefully construct a graph of the function showing how it passes through the points where
y = -1, y = 0, y = 1 and how it approaches the vertical asymptotes.

8.      Suppose sin t = 16/65 and t is in the first quadrant.  Find the following:

a.        

b.       

c.      

9.      Complete the table of values for , plot the points and sketch a graph.
 

 

10.  The Millennium Wheel rotates once every 30 minutes.  Its highest point is about 135 meters above the ground and the lowest point is about 5 meters above the ground.  Write a function that gives the height of a rider t minutes after boarding the Millennium Wheel.

Math 5  Trigonometry  Chapter 4 Test Solutions  fall ’07

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

a.       Find the reference number for t.
ANS:  so the reference number is .

b.      Find the coordinates of the terminal point P(x,y).
ANS: Since this point is in the third quadrant, both x and y are negative and so  and .

c.       Illustrate this point’s position on a plot of the unit circle.
ANS:  The point  

 

  1. Consider the point  

a.       Verify that the point lies on the unit circle. 
ANS: 

b.      Use the diagram at right to approximate to the nearest tenth a value of t  so that
ANS:  A vertical segment is drawn from 0.47 on the x-axis intersects the circle at t near 1.1

c.       Approximate to the nearest tenth a value of t so that
ANS: Since cot(t) = cos(t)/sin(t) = 8/15 and tan(π/2  t) = cot(t).  So choose t = 1.6  1.1 = 0.5

3.      Suppose that  and  point and sin(t) < 0. 
Find  
ANS: 
Thus   

4.      Write  in terms of , assuming the terminal point for t is in quadrant III.
ANS: Starting with , divide through by  to obtain .  Since sec(t) is negative in quadrant III,  

5.      Find the amplitude, period and phase shift of , construct a table of values and graph one period of the function, clearly showing the position of key points.|

ANS:  The amplitude is 4,
           the period is 1
and the phase angle is 1/4.

Graph is shown at right.

                                             

6.      Find an equation for the sinusoid whose graph is shown:

ANS:  The lowest point is at y=7 and the highest point is at 23 so the line of equilibrium is at the average of these: y = (7+23)/2 = 15. and the amplitude is (23  7)/2 = 8.  

The two peaks shown in the graph here are where x = 0.5 and x = 2.5, so the period is 2.5  0.5 = 2.
Thus an equation for the sinusoid is .

7.      Consider the function .

a.       Find the equations for two adjacent vertical asymptotes and sketch them in with dashed lines. 
ANS: We want the input to the tangent to be , that is  

b.      Find the x-coords where y = 0 and where .
ANS:  We want to find where the input to the tangent function is equal to , that is .

c.       Graph of the function showing how it passes through the points where y = -1, y = 0, y = 1 and how it approaches the vertical asymptotes.

8.      Suppose sin t = 16/65 and t is in the first quadrant.  Find the following:

a.        

b.       

c.        



  1. Complete the table of values for , plot the points and sketch a graph.
     

 

 
  1. The Millennium Wheel rotates once every 30 minutes.  Its highest point is about 135 meters above the ground and the lowest point is about 5 meters above the ground.  Write a function that gives the height of a rider t minutes after boarding the Millennium Wheel.
    ANS: