$\sin{(90^\circ+\theta)}$ identity

Formula

$\sin{(90^\circ+\theta)} \,=\, \cos{\theta}$

Introduction

Let theta be an angle in sexagesimal system, which means the symbol theta represents an angle in degrees. The sum of angles ninety degrees and theta is written as $90^\circ+\theta$.

The sine of the sum of angles ninety degrees and theta is written in mathematical form as follows.

$\sin{(90^\circ+\theta)}$

The sine of the compound angle ninety degrees plus theta is equal to the value of cosine of angle theta.

$\sin{(90^\circ+\theta)}$ $\,=\,$ $\cos{\theta}$

Usage

It is used as a formula in trigonometry to convert the sine of a compound angle ninety degrees plus an angle in terms of cosine of angle.

Example

Evaluate $\sin{135^\circ}$

The value of sine of angle one hundred thirty five degrees is not known to us but it can be evaluated easily by the sine of ninety degrees plus angle theta formula.

$\implies$ $\sin{135^\circ}$ $\,=\,$ $\sin{(90^\circ+45^\circ)}$

$\implies$ $\sin{(90^\circ+45^\circ)}$ $\,=\,$ $\cos{45^\circ}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sin{(90^\circ+45^\circ)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$

Thus, the sine of angle ninety degrees plus theta identity is used to find the sine of any angle in the second quadrant by transforming it into cosine of an angle in the first quadrant.

Other form

The sine of ninety degrees plus angle theta trigonometric identity is also popularly written in the following form as per the circular system.

$\sin{\Big(\dfrac{\pi}{2}+x\Big)} \,=\, \cos{x}$

Proofs

The sine of sum of the angles ninety degrees and theta trigonometric identity can be derived in two different mathematical approaches. So, let’s learn how to prove sine of ninety degrees plus theta formula in each method.

Trigonometric method

Learn how to derive the sine of angle ninety degrees plus theta identity by the trigonometric identities.

Geometric method

Learn how to prove the sine of angle ninety degrees plus theta formula by the geometric approach.