Math 5 Trigonometry  Final Exam Solutions  Fall ’06

1.      A right triangle with legs 8 and 15 is inscribed in a circle centered at the origin, as shown in the diagram.

a.       The area of the shaded region is  

b.      To find the coordinates of P, observe the similar triangles  so the y coord. of P is found by  and so  

2.      To find an equation for the line tangent to the circle  at (5, 12) note that the radius to that point has slope  so the perpendicular radius is , whence the point-slope formula yields the equation

3.      To find an equation for the line perpendicular to the line segment from (0,8) to (10,0) and passing through its center simply plug into slope intercept equation: .

4.      Given the plot of  shown,

a.        is a shift 4 up and 2 to the right.  The order of transformations doesn’t matter.

5.                   Can be thought of as either a vertical stretch by 2 and a shift 4 up followed by a horizontal shrink by 2 and a shift 3 left,
                          or
a shift of 2 up and a vertical stretch by 2 followed by a shift of 6 left and a horizontal shrink by 2.

6.     



7.      A sketch for the function
 is shown at right.  The dotted lines show how each piece would be continued if its domain was not limited.

8.      If the point P is on the unit circle is in QII and has
y =  then  

 

9.       has amplitude = , period = 6π and phase shift = . As shown: 



10.  Consider  

a.       The domain of the function is  

b.      The range of the function is  

c.       Sketch a graph of the function showing one period.  Remember to scale and label axes.

11.  If a potter’s wheel with radius 6 inches spins at  180 rpm then the angular speed of a point on the rim of the wheel is  and the linear speeds is .

12.  In the figure at right, we find x =  and
y =  


13.  From a point A on the ground, the angle of elevation to the top of a tree is 28.1°.  From a point B, 17 feet closer, the angle of elevation is measured to be 34.2°.  To find the height H of the tree, Let x = OB in the figure below and note that  while  so that  whence  feet.
                        

 

14.  Write the conic in standard form and sketch a graph indicating key features:

a.        is an ellipse centered at , has major axis of length  from  to  and minor axis of length  from  to  .  The foci are at

b.       is a parametric description of the hyperbola .  Vertices are at , foci at  and asymptotes along .

 

 

1.      The diagram shows circular quadrilateral ABCD with diagonals AB and CD.

a.       We show that if we construct M as shown so that , then
Since  , that makes two congruent angles for  so the result follows.

b.      Since adding equals to equals makes equals, , giving two congruent angles for  lead to  

c.       Conclude that