**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 *in ^{2}*; the volume of a sphere of radius 5

*in*is given by

*in*

^{3}.*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__

- sin (A)=
- cos(C)=
- csc(A)=
- tan(A)=

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

- 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.*