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Trigonometry Self Quiz--Solutions

1. Find exactly:
a.
b. tan
c. cos
d.

Solutions

a. =
b. tan = -1
c. cos =
d.  is undefined, as

If you had difficulty with this exercise, review the tutorials Right-Triangle Trigonometry and Trigonometric Functions.

2. If and , where is an angle in standard position:

a. In what quadrant is located?
b. Use reference triangles to find .
c. Compute .

Solutions

a.  Both the sine and the cosine functions take on positive values, so the angle is located in the first quadrant.
b.  In the diagram below, we see .

c.

If you had difficulty with this exercise, review the tutorials Trigonometric Functions and Trigonometric Identities.

3. Consider the following statement: “For some real angle , .” Is this
possible? Explain why or why not.

Solution
In the reference-triangles definition, we have where x is the x-coordinate of a point on the terminal side of the angle, and r is the distance from that point to the origin. The line segment of length r is the hypotenuse of the reference triangle, while another side of the reference triangle has length x; thus x<r. Therefore,  for all real angles .  Hence,  has no solution.

If you had difficulty with this exercise, review the tutorial Trigonometric Functions.

4.  Give at least two definitions of , where is a real number.

Solutions
Possibilities include , , or , where x is the x-coordinate of a point on the terminal side of  and y is the y-coordinate of a point on the terminal side of .

If you had difficulty with this exercise, review the tutorial Trigonometric Functions.

5. If a sphere has radius 5 inches, find its surface area and volume.

Solution
The surface area of a sphere of radius 5 in is given by  in2; the volume of a sphere of radius 5 in is given by in3.
If you had difficulty with this exercise, review the tutorial Formulas From Geometry.

6. Is the point  on the graph of  some real number k? If it is, give a possible value for k; if not, explain why.

Solution
The point is not on the graph. The function  is undefined at . Notice that the graph of  has a vertical asymptote at .

If you had difficulty with this exercise, review the tutorial Graphs of Trigonometric Functions.

7. Given the right triangle below, find

a. sin (A)

b. cos(C)

c. csc(A)

d. tan(A)

Solutions

1. sin (A)=
2. cos(C)=
3. csc(A)=
4. tan(A)=

If you had difficulty with this exercise, review the tutorial Right-Triangle Trigonometry.

1. Suppose the angle of elevation of the sun is 26.4°.  Find the length of the shadow cast by a 16-foot flagpole.

Solution

The flagpole is the vertical leg of a right triangle whose horizontal leg of length S is the shadow, with the 26.4° angle containing the horizontal leg and the hypotenuse.  Using trigonometry, tan(26.4°) = .  Thus, S =  32.2 ft.

If you had difficulty with this exercise, review the tutorial Real-World Applications of Trigonometry.