Question

Solving a Right Triangle Solve the right triangle.

Short answer

The sides of the given right-angled triangle with respect to the angle$\mathit{\theta }\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{6}}$ are

opposite side$\mathbf{=}\mathbf{41}\mathbf{.74}$

adjacent side$\mathbf{=}\mathbf{72}\mathbf{.3}$

hypotenuse$\mathbf{=}\mathbf{83}\mathbf{.48}$

The unknown angle of the triangle is${\mathbf{60}}^{\mathbf{°}}$

Step-by-step solution

Step 1.Given information.

The given angle is$\theta =\frac{\pi }{6}$ and hypotenuse$83.48$.

Step 2. Applying the trigonometric ratios in the right-angled triangle of the given figure to find the unknown sides.

Substituting the known values of and the hypotenuse in the above trigonometric ratio formulae,

$\begin{array}{l}\tan \theta =\frac{opp}{adj}\\ \tan \left(\frac{\pi }{6}\right)=\frac{x}{72.3}\\ \Rightarrow x=41.74\end{array}$

Similarly, to compute the hypotenuse,

$\begin{array}{c}\sin \theta =\frac{opp}{hyp}\\ \Rightarrow \sin \left(\frac{\pi }{6}\right)=\frac{41.74}{y}\\ \Rightarrow y=\frac{41.74}{0.5}\\ \Rightarrow y=83.48\end{array}$

Step3. Calculating angle

Calculating $\theta$of the given triangle, we get,

$\begin{array}{c}\frac{\pi }{6}+\frac{\pi }{2}+\theta =\pi \\ \Rightarrow \theta =\frac{\pi }{3}\end{array}$