Question

Solving a Right Triangle Solve the right triangle.

Short answer

The sides of the given right-angled triangle with respect to the angle are:

$\begin{array}{l}\mathbf{\angle }\mathbf{B}\mathbf{=}{\mathbf{54}}^{\mathbf{°}}\\ \mathbf{C}\mathbf{=}\mathbf{180}\mathbf{.33}\\ \mathbf{b}\mathbf{=}\mathbf{145}\mathbf{.89}\end{array}$

Step-by-step solution

Step 1.Given information.

The angle is given as:$\frac{\pi }{5}$

The length of the side is given as 106.

Step 2. Solve the triangle:

Since angle $A=\frac{\pi }{5}={36}^{°}$, then angle

$\begin{array}{l}A=\frac{\pi }{5}={36}^{°}\\ \Rightarrow \angle B={90}^{°}-{36}^{°}\\ ={54}^{°}\\ B={54}^{°}\\ \frac{\text{forside}c}{\sin {36}^{°}}=-\frac{\text{opposite}\left(BC\right)}{\text{hypotenuse}\left(BA\right)}\\ =\frac{a}{c}\end{array}$

$\begin{array}{l}\Rightarrow \text{(hypotenuse)}BA=\frac{106}{\sin 36}\\ =\frac{106}{0.587}\\ =180.33C\\ =180.33\end{array}$

Step 3. Find the value of b :

$\begin{array}{l}\therefore \frac{\text{forside}b}{\cos 36}=\frac{\text{adjecent}\left(CA\right)}{\text{hypotenuse}\left(BA\right)}\\ =\frac{b}{c}\\ \Rightarrow b=\cos 36\times c\\ \Rightarrow b=0.809\times 180.33\\ \Rightarrow b=145.89\end{array}$