Fundamental trigonometric relations

Example

Knowing that $\cos (x)=\sin (x)$ and $\sin (y)=2\cos (y)$, we are going to calculate the cosine and the sine of $x+y$:

$\begin{array}{rl}\sin (x+y)=& \sin (x)\cos (y)+\cos (x)\sin (y)=\sin (x)\cos (y)+2\sin (x)\cos (y)\\ =& 3\sin (x)\cos (y)\end{array}$

$\begin{array}{rl}\cos (x+y)=& \cos (x)\cos (y)-\sin (x)\sin (y)=\sin (x)\cos (y)-2\sin (x)\cos (y)\\ =& -\sin (x)\cos (y)\end{array}$

Proceeding in a similar way, we obtain that the sine and the cosine of $x-y$ is:

$\sin (x-y)=-\sin (x)\cos (y)=\cos (x+y)$

$\cos (x-y)=3\sin (x)\cos (y)=\sin (x+y)$

Example

Knowing that $\sin \alpha ={\displaystyle \frac{3}{5}}$, and that ${90}^{\circ }<\alpha <{180}^{\circ }$, calculate the remaining trigonometric ratios of the angle $\alpha$.

We are going to calculate other trigonometric ratios from the fact that we know the sine of the angle,

$$\sin (\alpha )={\displaystyle \frac{3}{5}}\Rightarrow \csc (\alpha )={\displaystyle \frac{5}{3}}$$

On the other hand, the cosine is: $$\cos (\alpha )=-\sqrt{1-{\sin }^2(\alpha )}=-\sqrt{1-{\displaystyle \frac{9}{25}}}=-\sqrt{{\displaystyle \frac{26}{25}}}=-{\displaystyle \frac{4}{5}}\Rightarrow \sec (\alpha )=-{\displaystyle \frac{5}{4}}$$

Finally, to calculate the tangent we use:

$$\tan (\alpha )={\displaystyle \frac{\sin (\alpha )}{\cos (\alpha )}}=-{\displaystyle \frac{{\displaystyle \frac{3}{5}}}{{\displaystyle \frac{4}{5}}}}=-{\displaystyle \frac{3}{4}}\Rightarrow \cot (\alpha )=-{\displaystyle \frac{4}{3}}$$