Question

Solving a Right

Triangle Solve the right triangle

Short answer

Perpendicular of triangle is $\mathbf{44}\mathbf{.797}$, the hypotenuse is $\mathbf{56}\mathbf{.849}$and the remaining angle is${\mathbf{38}}^{\mathbf{\circ }}$

Step-by-step solution

Step 1. Given information

A right triangle with known value of a side of 35 and an angle of${52}^{\circ }$

Step 2. Concept used

The ideas of a side and an angle are well-known. Use trigonometric equations to get the unknown side. Use the triangle sum property of angles to find an unknown angle. 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

Now use the formulae,

$\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan 52=\frac{p}{35}\\ 1.279=\frac{p}{35}\\ p=35\times 1.279\\ p=44.797\end{array}$

Use sum of triangle’s angle,

$\begin{array}{l}\angle A+\angle B+\angle C={180}^{\circ }\\ \angle B+52+90=180\\ \angle B=180-142\\ \angle B={38}^{\circ }\end{array}$

The remaining side,

$\begin{array}{l}\cos \theta =\frac{b}{h}\\ \cos 52=\frac{35}{h}\\ 0.615=\frac{35}{h}\\ h=\frac{35}{0.615}\\ h=56.849\end{array}$