## Trigonometric Function

Subject: Optional Mathematics

#### Overview

Different standard angles are taken as a variable in the trigonometric function. For different angles, the values of trigonometric ratios will also be different. Take the angles of x-axis and values of trigonometric ratios in y-axis Join the points freely to obtain the graph of the trigonometrical ratio.

### Graph of Trigonometric Function

Different standard angles are taken as a variable in the trigonometric function. For different angles, the values of trigonometric ratios will also be different. Take the angles of x-axis and values of trigonometric ratios in the y-axis. Join the points freely to obtain the graph of the trigonometrical ratio. For example: To draw the graph of y = sin x, we take x as standard angles 0o , 30o, 60o, 90o, 120o, 150o, 180o, 210o, 240o, 270o, 300o, 330o, 360o.

Some standard values of x and the corresponding values of sin x are given below:

 x y = sinx 0° 0 30° 0.5 45° 0.71 60° 0.87 90° 1 120° 0.87 135° 0.71 150° 0.5 180° 0 210° -0.5 225° -0.71 240° -0.87 270° -1 300° -0.87 315° -0.71 330° -0.5 360° 0

Plotting these values in the graph paper, we get the following graph. We get,

##### Things to remember
 x y = sinx 0° 0 30° 0.5 45° 0.71 60° 0.87 90° 1 120° 0.87 135° 0.71 150° 0.5 180° 0 210° -0.5 225° -0.71 240° -0.87 270° -1 300° -0.87 315° -0.71 330° -0.5 360° 0
• It includes every relationship which established among the people.
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• common interests and common objectives are not necessary for society.
##### More trig graphs

soln: From the equation different values of y = sin3x:

 x y=sin3x 0° 0 10°$\frac{π}{18}$ $\frac{1}{2}$(=0.5) 20°=$\frac{π}{9}$ $\frac{\sqrt{3}}{2}$(=0.87) 30°=$\frac{π}{6}$ 1 40°=$\frac{2π}{9}$ $\frac{\sqrt{3}}{2}$(=0.87) 50°=$\frac{\sqrt{5π}}{18}$ $\frac{1}{2}$(=0.5) 60°=$\frac{π}{3}$ 0
 70°=$\frac{7π}{18}$ -$\frac{1}{2}$(=-0.5) 80°=$\frac{4π}{9}$ -$\frac{\sqrt{3}}{2}$=-0.87 100°=$\frac{5π}{9}$ -1 110°=$\frac{11π}{18}$ -$\frac{1}{2}$=-0.50 120°=$\frac{2π}{3}$ 0
 150°$\frac{5π}{6}$ 1 180°=π 0

graph of above table:

soln; Different values of y = sin2θ

 θ y=sin2θ 0° 30°$\frac{π}{6}$ 60° 90°=$\frac{π}{2}$ 120°=$\frac{2π}{3}$ 150°=$\frac{5π}{6}$ 180°=π 0 $\begin{pmatrix} 1 \\ 2 \\ \end{pmatrix}$2=$\frac{1}{4}$(=0.25) $\frac{\sqrt{3}}{2}$2(=0.75) 1 $\frac{\sqrt{3}}{2}$2=0.75 $\begin{pmatrix} 1 \\ 2 \\ \end{pmatrix}$2=$\frac{1}{4}$=0.25 0=0

The graph o f the above table:

soln: Different values of y = cos2x

 θ y=cos2x 0 30°=$\frac{π}{6}$ 60°=$\frac{π}{3}$ 90°=$\frac{π}{2}$ 120°=$\frac{2π}{3}$ 150°=$\frac{5π}{6}$ 180°=π 0 $\frac{1}{2}$=0.5 $\frac{-1}{2}$=-0.5 -1 $\begin{pmatrix} -1 \\ 2 \\ \end{pmatrix}$=-0.5 $\frac{1}{2}$=0.5 1

The graph of above table:

Soln: Different values of y = cosθ - sinθ

 θ y=cosθ-sinθ 0° 30°=$\frac{π}{6}$ 60°=$\frac{π}{3}$ 90°=$\frac{π}{2}$ 120°=$\frac{2π}{3}$ 150°=$\frac{5π}{6}$ 180°=$\frac{π}{2}$ 1 $\frac{\sqrt{3}}{2}$-$\frac{1}{2}$=0.87-0.5=0.37 $\frac{1}{2}$-$\frac{\sqrt{3}}{2}$=0.5-0.87=-0.37 0-1=-1 -$\frac{1}{2}$-$\frac{\sqrt{3}}{2}$ = -0.5-0.87=-1.37 -$\frac{\sqrt{3}}{2}$-$\frac{1}{2}$=-0.87-0.5=-1.37 -1-0=1

The graph of above table:

soln: Different values of y = 2sinθ

 θ y=2sinθ 0° 30°=$\frac{π}{6}$ 60°=$\frac{π}{3}$ 90°=$\frac{π}{2}$ 120°=$\frac{2π}{3}$ 150°=$\frac{5π}{6}$ 180°=$\frac{π}{2}$ 0 1 $\sqrt{3}$=1.73 2 $\sqrt{3}$=1.73 1 0

The graph of the above table:

soln: Different values of y = tan2θ

 θ y=tan2θ 0° 30°=$\frac{π}{6}$ 60°=$\frac{π}{3}$ 90°=$\frac{π}{2}$ 120°=$\frac{2π}{3}$ 150°=$\frac{5π}{6}$ 180°=π 0 $\sqrt{3}$=1.73 -$\sqrt{3}$=-1.73 0 $\sqrt{3}$=1.73 -$\sqrt{3}$=-1.73 0

The graph of above table:

soln: Different values of y = tan$\frac{3x}{2}$

 θ y=tan$\frac{3x}{2}$ 0° 30°=$\frac{π}{6}$ 60°=$\frac{π}{3}$ 90°=$\frac{π}{2}$ 120°=$\frac{2π}{3}$ 150°=$\frac{5π}{6}$ 180°=π 0 1 ∞ -1 0 1 ∞

the grape of above table :