Exercises: Sine, Cosine and Tangent on a right triangle

Given: $a=7\text{cm}$; $b=18\text{cm}$

Required: $\alpha ,\beta ,$

You may start by drawing a sketch.

Compute $\alpha$ using the tangent.

$\tan \left(\alpha \right)=\frac{a}{b}$

Insert the values and calculate $\alpha$ using the calculator.

$\tan (\alpha )=\frac{7\text{cm}}{18\text{cm}}$

$\alpha ={21.3}^{\circ }$

Compute $\beta$. $\alpha$ and $\beta$ add up to 90° , so for $\beta$, we have:

$\beta ={90}^{\circ }-{\mathrm{21,3}}^{\circ }={68.7}^{\circ }$

$\Rightarrow$ The angle $\alpha$ is 21.3°. $\beta$ equals 68.7°.

Given: $\alpha ={68.7}^{\circ };\beta ={21.3}^{\circ }$

Required: $\gamma ,\delta$

You may start by drawing a sketch.

$\delta$ equals 2 $\alpha$ , since $\delta$ $+$ $\gamma$ $=$180° and $\gamma$ $+2$ $\alpha$ $=$180° holds in a right triangle.

$\Rightarrow$ $\delta$ + $\gamma$ $=$ $\gamma$ $+2$ $\alpha$

$\Rightarrow$ $\delta$ $=$$2$ $\alpha$

$\delta =2\cdot {68.7}^{\circ }={137.4}^{\circ }$

$\gamma$ and $\delta$ add up to 180°.

$\gamma ={180}^{\circ }-{137.4}^{\circ }={42.6}^{\circ }$

$\Rightarrow$ The angle  $\gamma$ equals 42,6°.  $\delta$ amounts to 137.4°.