Math Doubts

Sin squared formula

Expansion form

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

Simplified form

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

How to use

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

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

Proof

The sine squared formula is derived from the Pythagorean identity of sin and cos functions.

If angle of a right triangle is theta, then the sum of squares of sin and cos functions equals to one.

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

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

Therefore, it is proved that sine squared theta equals to subtraction of cos squared theta from one.

Alternative form

The sine squared identity is also written in terms of different angles.

For example, if angle of right triangle is denoted by $x$, then the sine squared formula is written as $\sin^2{x} \,=\, 1-\cos^2{x}$

Remember, the angle of right angled triangle can be denoted by any symbol, the sine squared formula should be written in terms of the respective symbol.



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