Math Doubts

Sin squared formula

Formula

$\sin^2{\theta} \,=\, 1-\cos^2{\theta}$

The square of sine function equals to the subtraction of square of cos function from one is called the sine squared formula. It is also called as the square of sin function identity.

Introduction

The sine functions are often appeared in square form in trigonometric expressions and equations. They can be simplified by transforming the square of sine functions into its equivalent form. So, it is essential to learn the square of sine function identity for studying the trigonometry further.

Usage

The sine squared trigonometric identity is used as a formula in two cases.

  1. The square of sin function is expanded as the subtraction of cos squared function from one.
  2. The subtraction of cos squared function from one is simplified as the square of sin function.

Popular forms

The square of sine function rule is also expressed in two popular forms in mathematics.

  1. $\sin^2{x} \,=\, 1-\cos^2{x}$
  2. $\sin^2{A} \,=\, 1-\cos^2{A}$

In this way, you can write the square of sine function rule in terms of any angle in mathematics.

Proof

Let theta be an angle of a right triangle, then the sine and cosine are written in mathematical form as $\sin{\theta}$ and $\cos{\theta}$ respectively. According to the Pythagorean identity of sin and cos functions, the relationship between sine and cosine can be written in the following mathematical form.

$\sin^2{\theta}+\cos^2{\theta} \,=\, 1$

$\therefore \,\,\,\,\,\, \sin^2{\theta} \,=\, 1-\cos^2{\theta}$

Therefore, it is proved that the square of sine function is equal to the subtraction of the square of cosine function from one.

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