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

Formula

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

Introduction

Let’s assume that a Greek alphabet Theta is considered to denote an angle in first quadrant of the coordinate system. The right angle is written as ninety degrees as per the sexagesimal system in mathematics. Now, subtract the angle theta from the right angle and it is written as $90^\circ-\theta$ mathematically. The difference of them obviously represents an angle in the first quadrant of the coordinate system.

The sine of that angle is written as $\sin{(90^\circ-\theta)}$ and its value is mathematically equal to cosine of angle theta.

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

It is called the cofunction identity of sine in trigonometry.

Other form

The sine of ninety degrees minus theta trigonometric identity is also alternatively written as follows.

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

Uses

The sine of ninety degrees minus theta identity is mainly used in two cases in trigonometry.

1. It is used to simplify the sine of ninety degrees minus theta function as the cosine of angle theta.
2. It is used to calculate the sine of angle in first quadrant by the cosine of the complementary angle.

Proofs

The sine of angle ninety degrees minus theta identity can be derived in two distinct methods in mathematics.

Trigonometric Method

Learn how to derive the sine ninety minus theta formula in trigonometry by a trigonometric identity.

Geometric Method

Learn how to prove the sine ninety minus theta identity geometrically by constructing a right triangle in first quadrant of coordinate system.

Verification

Assume that $\theta \,=\, 30^\circ$.

Now, let’s evaluate both sine of ninety degrees minus theta and cosine of angle theta by replacing the theta with thirty degrees.

$(1).\,\,\,$ $\sin{(90^\circ-30^\circ)}$ $\,=\,$ $\sin{(60^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}}{2}$

$(2).\,\,\,$ $\cos{(30^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}}{2}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sin{(90^\circ-30^\circ)}$ $\,=\,$ $\cos{(30^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}}{2}$

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