Question

$\maltese$ Derive the following trigonometric functions :-


• sin (-x) = - sin (x)

• cos (-x) = - cos (x)​

Answer 1

Topic: Trigonometry(Convention)

Convention

The angle of a negative value is not supported. Let's make a new convention to calculate those values.

Let the positive x-axis be the initial line.

The circle centered in the origin and having the radius of 1 unit is called the unit circle. For $P(x,y)$ on the unit circle forming the angle of $\theta$ with the initial line,

• $\cos\theta=\dfrac{x}{1} =x$
• $\sin\theta=\dfrac{y}{1}=y$
• $\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{x}{y}$

This is the convention that allows us to calculate trigonometric values of angles other than acute angles.

Solution

Let the line of symmetry be the x-axis.

Then let's reflect $P(x,y)$. $P'(x,-y)$ forms a symmetry against the x-axis.

According to the convention $\sin(-\theta)=\dfrac{-y}{1}=-\sin\theta$ as well as $\cos(-\theta)=\dfrac{x}{1} =\cos\theta$.

More information

A function in which the graph is symmetric against the y-axis is called an even function. A function in which the graph is symmetric against the origin is called an odd function.

Since $\sin (-x)=-\sin x$ it is called an odd function, similarly $\cos(-x)=\cos x$ so it is an even function.

$\tan(-\theta)=\dfrac{-y}{x} =-\tan\theta$, so $\tan (-x)=-\tan x$. It is an odd function.

Answer 2

Answer:

hope the above image helps you