Trigonometric identities in one angle

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

1. $\sin^2 \alpha +\cos ^2 \alpha =1$
2. $\displaystyle 1+\tan^2 \alpha=\frac{1}{\cos ^2\alpha}= \sec^2 \alpha$

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

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

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

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