Question

In a right triangle, θ​ is an acute angle and tanθ=6/5 What is the exact value of cosθ​

Answer 1

Recall the Pythagorean identity,

`\tan^2\theta+1=\sec^2\theta`

`\sec\theta` is the reciprocal of
`\cos\theta`. In a right triangle, the non-right angles are always acute, so
`\cos\theta` would be positive for either of them. This means

`\frac1{\cos^2\theta}=\tan^2\theta+1\implies\cos^2\theta=\frac1{\tan^2\theta+1}\implies\cos\theta=\frac1{√(\tan^2\theta+1)}`

We have
`\tan\theta=\frac65`, so

`\cos\theta=\frac1{√(\left(\frac65\right)^2+1)}=\frac5{√(61)}`