Cos of negative angle

A mathematical relation of cosine of negative angle with cosine of positive angle is called cosine of negative angle identity.

Formula

$\cos{(-\theta)} \,=\, \cos{\theta}$

Proof

The mathematical relation between cosine of negative angle and cosine of positive angle is derived in mathematical form in trigonometry by geometrical method.

Construction of triangle with positive angle

positive angle

$\Delta QOP$ is a right angled triangle, constructed with an angle theta. Write cosine of angle in terms of ratio of lengths of the sides.

$\cos{\theta} \,=\, \dfrac{OQ}{OP}$

$\Delta QOP$ is constructed in first quadrant. So, the lengths of adjacent and opposite sides are positive. Hence, take the lengths of both sides as $x$ and $y$ respectively.

$\implies$ $\cos{\theta} \,=\, \dfrac{x}{\sqrt{x^2+y^2}}$

Construction of triangle with negative angle

positive angle

Construct same triangle but with negative angle. So, the angle of the $\Delta ROQ$ will be $–\theta$.

Express, the trigonometric ratio cosine in terms of ratio of the lengths of the associated sides.

$\cos{(-\theta)} \,=\, \dfrac{OQ}{OR}$

In this case, the length of opposite side will be $–y$ but the length of adjacent side is $x$ geometrically.

$\implies$ $\cos{(-\theta)} \,=\, \dfrac{x}{\sqrt{x^2+y^2}}$

Comparing Cosine functions

Compare equations of both cosine of positive angle and cosine of negative angle.

$\cos{\theta} \,=\, \dfrac{x}{\sqrt{x^2+y^2}}$

$\cos{(-\theta)} \,=\, \dfrac{x}{\sqrt{x^2+y^2}}$

The two equations reveals that cosine of negative angle is always equal to cosine of angle.

$\,\,\, \therefore \,\,\,\,\,\,$ $\cos{(-\theta)} \,=\, \cos{\theta}$

It is called cosine of negative angle identity and often used as a formula in trigonometric mathematics.



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