Cofunction Identities

A step-by-step guide to cofunction 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.

Cofunction identities in radians

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

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:

\(sin\:x=cos\:40°\)

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

\(90°-\:x=40°\)

\(x=90°-40°\)

\(x=50°\)