If cos θ=45, find all other trigonometric ratios of angle θ.
Question:

If $\cos \theta=\frac{4}{5}$, find all other trigonometric ratios of angle $\theta$.

Solution:

Given: $\cos \theta=\frac{4}{5}$

Now, we have to find all the other trigonometric ratios.

We have the following right angle triangle.

From the above figure,

Perpendicular $=\sqrt{\text { Hypotenuse }^{2}-\text { Base }^{2}}$

$\Rightarrow A B=\sqrt{A C^{2}-B C^{2}}$

$\Rightarrow A B=\sqrt{5^{2}-4^{2}}$

$\Rightarrow A B=3$

Therefore,

$\sin \theta=\frac{A B}{A C}=\frac{3}{5}$

$\operatorname{cosec} \theta=\frac{A C}{A B}=\frac{5}{3}$

$\sec \theta=\frac{A C}{B C}=\frac{5}{4}$

$\tan \theta=\frac{A B}{B C}=\frac{3}{4}$

$\cot \theta=\frac{B C}{A B}=\frac{4}{3}$