Question

I need help to find the missing side lengths. I will include the photo

Answer 1

The right angled triangle has the reference angle given as 30 degrees. That means the side given is the OPPOSITE, side length a is the HYPOTENUSE and side length b is the ADJACENT.

Hence, we would have the following;

`\begin{gathered} \sin \theta=(opp)/(hyp) \\ \sin 30=\frac{\frac{4\sqrt[]{5}}{5}}{a} \\ \sin 30=(1)/(2) \\ \text{Therefore,} \\ (1)/(2)=\frac{4\sqrt[]{5}}{5a} \\ \text{Cross multiply and you have} \\ 5a=2*4\sqrt[]{5} \\ 5a=8\sqrt[]{5} \\ \text{Divide both sides by 5 and you have} \\ a=\frac{8\sqrt[]{5}}{5} \end{gathered}`

To calculate side length b

`\begin{gathered} \tan \theta=(opp)/(adj) \\ \tan 30=\frac{\frac{4\sqrt[]{5}}{5}}{b} \\ \tan 30=\frac{1}{\sqrt[]{3}} \\ \text{Therefore} \\ \frac{1}{\sqrt[]{3}}=\frac{4\sqrt[]{5}}{5b} \\ \text{Cross multiply and you'll have} \\ 5b=\sqrt[]{3}*4\sqrt[]{5} \\ 5b=4\sqrt[]{3*5} \\ 5b=4\sqrt[]{15} \\ \text{Divide both sides by 5} \\ b=\frac{4\sqrt[]{15}}{5} \end{gathered}`