# Trigonometric Graph Worksheet

Trigonometric Graph Worksheet
• Page 1
1.
Which of the following is a point on the curve $y$ = cot $x$?
 a. ($\frac{\pi }{6}$, $\frac{1}{\sqrt{3}}$) b. (0, 0) c. ($\frac{\pi }{3}$, $\frac{1}{\sqrt{3}}$) d. ($\frac{\pi }{2}$, 1)

#### Solution:

f(x) = cot x

Since f(0) = cot 0 is not defined ,

f(π6) = cot(π6) = 3 ,

f(π3) = cot(π3) = 13

f(π2) = cot(π2) = 0 ,

The graph of cot x passes through the points (π6, 3), (π3, 13) and (π2, 0)

So, the point (π3, 13) is on the curve y = cot x.

2.
The graph of cos $x$
 a. Passes through the origin b. Does not pass through ($\frac{\pi }{4}$, 0) c. Passes through the point ($\frac{\pi }{2}$, 1) d. Does not pass through the origin

#### Solution:

f(x) = cos x

Since f(0) = cos 0 = 1,

f( π2) = cos(π2) = 0

cos x is passing through (0, 1), (π2, 0).

So, graph of cos x does not pass through origin (0, 0).

3.
Which of the following is a point on the curve $y$ = tan $x$?
 a. (0, 1) b. (- $\frac{\pi }{2}$, 1) c. ($\frac{\pi }{2}$, 1) d. ($\frac{\pi }{4}$, 1)

#### Solution:

f(x) = tan x

Since f(0) = tan 0 = 0,

f( π4) = tan(π4) = 1

f( π2) = tan(π2) is not defined.

f(- π2) = tan(- π2) is not defined.

The curve passes through (0, 0), (π4, 1)

So, (π4, 1) is a point on y = tan x.

4.
Through which of the following points does the graph of sin $x$ pass?
 a. ($\frac{\pi }{6}$, $\frac{\sqrt{3}}{2}$) b. ($\frac{\pi }{6}$, $\frac{1}{2}$) c. ($\frac{\pi }{2}$, 0) d. (0, -1)

#### Solution:

f(x) = sin x

Since f(0) = sin 0 = 0 ,

f(π6) = sin(π6) = 12 and

f(π2) = sin π2 = 1.

The graph of sin x passes through the points (0, 0), (π6, 12) and (π2, 1).

5.
Which of the following is correct?
 a. The graph of sin $x$ passes through (0, 0) b. The graph of sin $x$ does not pass through the origin. c. The graph of sin $x$ does not pass through ($\frac{\pi }{2}$, 1) d. The graph of sin $x$ does not pass through (-$\frac{\pi }{2}$, -1)

#### Solution:

The graph of sinx passes through the points (0, 0), (π2, 1), (- π2, -1).

6.
The $y$ - intercept of the graph $y$ = sin $x$ is
 a. -1 b. 2 c. 1

#### Solution:

Substitute x = 0 in y = sin x to get its y- intercept.

y = sin 0 = 0
[Substitute x = 0.]

So, y-intercept of the graph y = sin x is 0.

7.
The zeros of the graph $y$ = cos $x$ are existing at $x$ =
 a. $\frac{n\pi }{2}$ for all integer values of $n$ b. $n$π for all integer values of $n$ c. (2$n$ + 1) $\frac{\pi }{2}$ for all real values of $n$ d. (2$n$ + 1) π for all integer values of $n$

#### Solution:

The zeros of y = sin x are the solutions of sin x = 0

The solutions of cos x = 0 are .....- 3π , - 2π , - π , 0, π , 2π , 3π , ..... that is the multiples of π which are in short n π for all integer values of n.

8.
The $y$ - intercept of $y$ = tan $x$ is:
 a. 2 b. -1 c. 1

#### Solution:

Substitute x = 0 in y = tan x to get its y-intecept.

y = tan 0 = 0
[Substitute x = 0.]

So, y-intercept of the graph y = tan x is 0.

9.
Write the amplitude of $y$ = 5sin($\frac{x}{4}$).
 a. $\frac{1}{3}$ b. 5 c. 4

#### Solution:

The amplitude of y = bsin(xa) is | b |
[Definition.]

On comparing y = 5sin (x4) with y = bsin (xa) we have b = 5, a = 4

So, the amplitude of y = 5in(x3) is | b | = | 5 | = 5

10.
Evaluate: $\underset{x\to 0}{\mathrm{lim}}$
 a. 23 b. $\frac{1}{9}$ c. $\frac{23}{9}$ d. $\frac{23}{3}$

#### Solution:

limx0 46x21-cos 6x
[Undefined at x = 0.]

= limx0 46x22sin2 3x
[Use 1 - cos 2x = 2sin2 x.]

= 23 limx0 1sin2 3xx2
[Divide both the numerator and the denominator by x2.]

= 23 1(limx0sin 3xx)2
[Use quotient law of limits.]

= 23 1(3)2 = 23 / 9
[Use limx0 sin kxx = k.]