# Odd and even Function Worksheet

Odd and even Function Worksheet
• Page 1
1.
What is the period of the function $y$ = - $\frac{1}{2}$tan 3$x$?
 a. 3$\pi$ b. $\frac{\pi }{6}$ c. $\frac{2\pi }{3}$ d. $\frac{\pi }{3}$

#### Solution:

y = - 1 / 2tan 3x
[y = a tan bx.]

Period is π3 or (180 / 3)°.
[πb or (180b)°.]

2.
Find the period of the function $y$ = tan 4$x$.
 a. 90o b. 45o c. 50o d. 720o

#### Solution:

y = tan 4x
[y = a tan bx.]

Period is π4 or (180 / 4
[πb or (180b)°.]

= π4 or 45°

3.
Which of the following is true for the function $y$ = cos $x$?
 a. Even function b. Odd function c. Odd and even function d. Cannot be determined

#### Solution:

If x is the measure of an angle with terminal side in quadrant I, then - x is the measure of an angle with terminal side in quadrant IV. Similarly, if x is the measure of a quadrant IV angle, then - x is the measure of a quadrant I angle. In either case, since cos x is negative in both quadrants I and II, cos(- x) = cos x.

If x is the measure of an angle with terminal side in quadrant II, then - x is the measure of an angle with terminal side in quadrant III. Also, if x is the measure of a quadrant III angle, then - x is the measure of a quadrant II angle. In either case, since cos x is negative in both quadrants II and III, cos(- x) = cos x.

Since cos (- x) = cos x for all values of x in the domain of the function, y = cos x is an even function.

4.
Which of the following is true for the function $y$ = csc $x$?
 a. Odd function b. Even function c. Odd and even function d. Cannot be determined

#### Solution:

If x is the measure of an angle with terminal side in quadrant I, then - x is the measure of an angle with terminal side in quadrant IV. Similarly, if x is the measure of a quadrant IV angle, then - x is the measure of a quadrant I angle. In either case, since csc x is negative in quadrant IV and positive in quadrant I, csc (- x) = - csc x.

If x is the measure of an angle with terminal side in quadrant II, then - x is the measure of an angle with terminal side in quadrant III. Also, if x is the measure of a quadrant III angle, then - x is the measure of a quadrant II angle. In either case, since csc x is negative in quadrant III and positive in quadrant II, csc (- x) = - csc x.

Since csc (- x) = - csc x for all values of x in the domain of the function, y = csc x is an odd function.

5.
Find the period of the function $y$ = 2 cot $x$.
 a. 2$\pi$ b. $\pi$ c. $\frac{\pi }{2}$ d. $\frac{2}{\pi }$

#### Solution:

y = 2 cot x
[y = a cot bx.]

Period is π1 or (180 / 1
[πb or (180b)°.]

= π or 180°

6.
Find the period of the function $y$ = $\frac{5}{3}$ cos ($\frac{x}{3}$).
 a. 60o b. 540o c. 300o d. 108o

#### Solution:

y = 5 / 3 cos (x3)
[y = a cos bx.]

Period is π13 or (18013
[π2 or (180b),°.]

= 3π or 540°

7.
What is the period of the function $y$ = - $\frac{3}{2}$sec 2$x$ ?
 a. 3$\pi$ b. $\frac{\pi }{2}$ c. $\pi$ d. 2$\pi$

#### Solution:

y = - 3 / 2sec 2x
[y = a sec bx.]

Period is 2π2 or (360 / 2)o
[2πb or (2 × 180b)o.]

= π or 180o

8.
Find the period of the function $y$ = sec 6$x$ + $\frac{3}{2}$.
 a. 90o b. 60o c. 30o d. 216o

#### Solution:

y = sec 6x + 3 / 2
[y = a sec bx + c.]

Period is 2π6 or (360 / 6)o
[2πb or (360b)o.]

= π3 or 60o

9.
Find the period of the function $y$ = 2 csc ($\frac{x}{2}$).
 a. $\pi$ b. 4$\pi$ c. $\frac{\pi }{2}$ d. 2$\pi$

#### Solution:

y = 2 csc (x2)
[y = a csc bx.]

Period is 2π12 or (36012)o
[2πb or (360b)o.]

= 4π or 720o

10.
Find the period of the function $y$ = - 3 csc ($\frac{2}{3}$)$x$.
 a. $\frac{3\pi }{4}$ b. 3$\pi$ c. $\frac{3\pi }{2}$ d. $\frac{4\pi }{3}$

#### Solution:

y = - 3 csc (2 / 3)x
[y = a csc bx.]

Period is 2π23 or (36023
[2πb or (360b)°.]

= 3π or 540°