Question

Finding an Unknown Side

Find the side labeled $\mathit{x}$.

Short answer

The value is$\mathit{x}\mathbf{=}\mathbf{16}\mathbf{.9705}$

Step-by-step solution

Step 1. Given information

A triangle with known value of a side of 12 and an angle of${45}^{\circ }$

Step 2. Concept used

In this question, by applying trigonometric formulae we can find unknown side of triangle. 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,

$\begin{array}{l}\Rightarrow \theta ={45}^{°}\\ \Rightarrow b=12\\ \Rightarrow h=x\end{array}$

Now use the formulae,

$\begin{array}{l}\cos \theta =\frac{b}{h}\\ \Rightarrow \cos 45=\frac{12}{x}\\ 0.70710678=\frac{12}{x}\\ \Rightarrow x=\frac{12}{0.70710678}\\ \Rightarrow x=16.9705628\\ \Rightarrow x\approx 16.9705\end{array}$