Question

Finding an Unknown Side

Find the side labeled. State your answer rounded to five decimal places.

Short answer

The value is$\mathit{x}\mathbf{=}\mathbf{6}\mathbf{.92820}$

Step-by-step solution

Step 1. Given information

A triangle with known value of a side of x and an angle of ${30}^{\circ }$

Step 2. Concept used

There is a side and an angle that are well-known. Use trigonometric equations to get the unknown side. Use formulae as-

$\begin{array}{l}\Rightarrow \sin \theta =\frac{p}{h}\\ \Rightarrow \cos \theta =\frac{b}{h}\\ \Rightarrow \tan \theta =\frac{p}{b}\end{array}$

Where

$\begin{array}{l}h=hypotenuse\\ b=base\\ p=perpendicular\end{array}$

Step 3. Calculation

From given figure,

$\begin{array}{l}\Rightarrow \theta ={30}^{°}\\ \Rightarrow b=x\\ \Rightarrow p=4\end{array}$

Now use the formulae,

$\begin{array}{l}\tan \theta =\frac{p}{b}\\ \Rightarrow \tan 30=\frac{4}{x}\\ \Rightarrow 0.5773502=\frac{4}{x}\\ \Rightarrow x=\frac{4}{0.5773502}\\ \Rightarrow x=6.9282032\\ \Rightarrow x\approx 6.92820\end{array}$