Trigonometric identities in one angle

Known some trigonometric ratio of one angle, we can easily calculate the rest through the following relationships:

$\sin^{2} α + \cos^{2} α = 1$
$1 + \tan^{2} α = \frac{1}{\cos^{2} α} = \sec^{2} α$

So, if we want to know the trigonometric ratios of one angle $α$ , we only need to know one of them and the quadrant where the angle is.

Let's suppose we have an angle $α$ and we know that $\sin α = \frac{1}{2}$ and that it belongs to the first quadrant, then it's quite easy to calculate its tangent and its cosine.

Example

We only need to do the following: $\sin^{2} α + \cos^{2} α = 1 \Rightarrow \frac{1}{4} + \cos^{2} α = 1 \Rightarrow \cos^{2} = \frac{3}{4} \Rightarrow$ $\Rightarrow \cos α = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$

$1 + \tan^{2} α = \frac{1}{\cos^{2} α} \Rightarrow \tan^{2} α = \frac{1}{\frac{3}{4}} - 1 = \frac{4}{3} - 1 = \frac{1}{3} \Rightarrow$ $\Rightarrow \tan α = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$