# Cofunction Identities

Cofunction identities show the relationship between the different trigonometric functions and their complementary angles. In this guide, you will learn more about cofunction identities.

## A step-by-step guide tocofunction identities

Cofunction identities are trigonometric identities that show a relationship between trigonometric functions and complementary angles.

We have six identities that can be obtained using right triangles, the angle sum property of a triangle, and trigonometric ratio formulas.

The cofunction identities establish a relationship between trigonometric functions $$sin$$ and $$cos$$, $$tan$$ and $$cot$$, and $$sec$$ and $$csc$$. These functions are known as cofunctions of each other.

We can write cofunction identities in terms of radians and degrees because these are the units of angle measurement.

• $$\color{blue}{sin\:\left(\frac{\pi }{2}\:-\:θ\right)=cos\:θ}$$
• $$\color{blue}{cos\:\left(\frac{\pi }{2}\:-\:θ\right)=sin\:θ}$$
• $$\color{blue}{tan\:\left(\frac{\pi }{2}-\:θ\right)=cot\:θ}$$
• $$\color{blue}{cot\:\left(\:\frac{\pi }{2}-θ\right)=tan\:θ}$$
• $$\color{blue}{sec\:\left(\frac{\pi }{2}-\:θ\right)=cosec\:θ}$$
• $$\color{blue}{csc\:\left(\frac{\pi }{2}-θ\right)=sec\:θ}$$

### Cofunction identities in degrees

• $$\color{blue}{sin\:\left(90°\:-\:θ\right)=cos\:θ}$$
• $$\color{blue}{cos\:\left(90°\:-\:θ\right)=sin\:θ}$$
• $$\color{blue}{tan\:\left(90°\:-\:θ\right)=cot\:θ}$$
• $$\color{blue}{cot\:\left(90°\:-\:θ\right)=tan\:θ}$$
• $$\color{blue}{sec\:\left(90°\:-\:θ\right)=cosec\:θ}$$
• $$\color{blue}{csc\:\left(90°-\:θ\right)=sec\:θ}$$

### Cofunction Identities – Example 1:

Find the value of acute angle $$x$$, if $$sin\:x=cos\:40°$$.

Solution

Using cofunction identity, $$cos\:\left(90°\:-\:θ\right)=sin\:θ$$, we can write $$sin\:x=cos\:40°$$ as:

$$sin\:x=cos\:40°$$

$$cos\:\left(90°-\:x\right)=cos\:40°$$

$$90°-\:x=40°$$

$$x=90°-40°$$

$$x=50°$$

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