Q27. - Algebra 1

Question

Find $\mathit{m}\mathbf{\angle }\mathit{X}$ to the nearest degree.

Step-by-Step Solution

Answer

Measure of angle $X$ to the nearest degree is given as,

 $\mathit{m}\mathbf{\angle }\mathit{X}\mathbf{=}\mathbf{37}\mathbf{°}$

Step 1. Define the trigonometric ratio ‘ sin θ ’.

If $\theta$ is any angle in a given right angle triangle, then the trignometric ratio  is given as,

 $\sin \theta =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}opposite\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle \theta }{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}$

Step 2. Calculate the measure of angle X ( m ∠ X ).

Observe the figure.

From the figure, $XZ$ is hypotenuse (side opposite to 90 degree) and length of $XZ$ is 15 units.

$YZ$ is side opposite to angle $X$ and the length of $YZ$ is 9 units.

 $\begin{array}{l}\sin X=\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}opposite\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle X}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ \sin X=\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}YZ}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}XZ}}\\ \sin X=\frac{9}{15}\\ \sin X=0.6\\ X={\sin }^{-1}\left(0.6\right)\\ X=36.87°\end{array}$

Step 3. State the conclusion.

The measure of angle $X$ is $36.87°$.

Here, $36.87°\approx 37° \left[\text{Rounding\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}the\hspace{0.17em}\hspace{0.17em}nearest\hspace{0.17em}\hspace{0.17em}degree}\right]$.

Therefore, measure of angle $X$$(m\angle X)$ to the nearest degree is $37°$.