Math Doubts

Cofunction identity of sin function

Formula

Sexagesimal System
$\sin{(90^\circ-\theta)} \,=\, \cos{\theta}$
Circular (or) Radian System
$\sin{\Big(\dfrac{\pi}{2}-x\Big)} \,=\, \cos{x}$

In this formula, the angle in degrees is denoted by Theta ($\theta$) and the angle in radians is denoted by $x$.

The angle of sin function is an allied angle in first quadrant. So, it is called as first quadrant’s allied angle identity of sin function. The angle of sin function is complement of the angle of the cos function. Therefore, it is also called as cofunction identity of sin function.

Proof

allied angle in first quadrant

Consider first quadrant. A line segment is rotated to some angle and it is named as $\overline{PQ}$.

Draw perpendicular lines to both horizontal and vertical lines from $Q$ and they interest them at points $S$ and $R$ respectively. Thus, it forms two right angled triangles $\Delta QPS$ and $\Delta QPR$ in first quadrant.

Take $\angle QPR = \theta$, then $\angle QPS = 90^\circ-\theta$.

Take $PS = g$, then $RQ = g$. Similarly, take $RP = h$, then $QS = h$.

Express sin of allied angle

allied angle in first quadrant

The angle of $\Delta QPS$ is an allied angle and it is $90^\circ-\theta$. Now, find the value of sine of first quadrant’s allied angle by expressing it in terms of ratio of the sides of the triangle.

$\sin{(90^\circ-\theta)} = \dfrac{SQ}{PQ}$

The length of the opposite side ($\overline{QS}$) is $h$ but the length of the hypotenuse is unknown. However, the length of the adjacent side is known. Therefore, the length of the hypotenuse can be calculated by Pythagorean theorem.

$\implies \sin{(90^\circ-\theta)} = \dfrac{h}{\sqrt{g^2+h^2}}$

Find equivalent value of the fraction

allied angle in first quadrant

In previous step, sin of allied angle of first quadrant is expressed as a fraction. Now, calculate the value of the fraction in the form a trigonometric function.

It can be done by considering $\Delta QPR$.

$\dfrac{h}{\sqrt{g^2+h^2}} = \dfrac{RP}{PQ}$

As per $\Delta QPR$, the angle of the triangle is theta. The ratio of lengths of $\overline{RP}$ to $\overline{PQ}$ is cos of angle theta.

$\implies \dfrac{h}{\sqrt{g^2+h^2}} = \cos{\theta}$

Cofunction identity of sin function

Geometrically, the two steps have proved that

$\sin{(90^\circ-\theta)}$ $=$ $\dfrac{h}{\sqrt{g^2+h^2}}$ $=$ $\cos{\theta}$

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

Sin of allied angle of first quadrant is equal to the cos of angle. So, it is known first quadrant’s allied angle identity of sin function. The two angles of both trigonometric functions are complementary angles. So, the trigonometric identity is called as cofunction identity of sin function.

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