Question

Solving a Right Triangle Solve the right triangle.

Short answer

The sides of the given right-angled triangle with respect to the angle${68}^{\circ }$

are

Opposite side=$\mathbf{927}\mathbf{.183}$

Adjacent side=$\mathbf{374}\mathbf{.606593}$

Hypotenuse=$\mathbf{1000}$

The unknown angle of the triangle is $\mathit{\theta }\mathbf{=}{\mathbf{22}}^{\mathbf{\circ }}$.

Step-by-step solution

Step 1.Given information.

A right-angled triangle with the given dimensions $\theta ={68}^{\circ }$and hypotenuse 1000.

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 $\theta ={68}^{\circ }$and the hypotenuse in the above trigonometric ratio formulae,

Step3. Calculation

$\begin{array}{l}sin\left({68}^{\circ }\right)=\frac{y}{1000}\\ 0.92718=\frac{y}{1000}\\ y=927.183\end{array}$

Similarly, to find the adjacent side,

$\begin{array}{l}\cos \left({68}^{\circ }\right)=\frac{x}{1000}\\ 0.3746065=\frac{x}{1000}\\ x=374.606593\end{array}$

Calculating $\theta$, we get,

$\begin{array}{l}{68}^{\circ }+{90}^{\circ }+\theta ={180}^{\circ }\\ {158}^{\circ }+\theta ={180}^{\circ }\\ \theta ={180}^{\circ }-{158}^{\circ }\\ \theta ={22}^{\circ }\end{array}$