Question

Write an equation involving $\sin ,\cos \text{or}\tan$that can be used to find x. Then solve the equation. Round measures of side to the nearest tenth and measures of angles to the nearest degrees.

Short answer

The equation is$\sin x{}^{\circ }=\frac{16}{22}$.And the simplified value of$x=47$.

Step-by-step solution

Step 1. Write down the given information.

The given figure is shown below.

Step 2. Concept used.

The six trigonometric functions are $\sin \theta , \csc \theta , \cos \theta , \sec \theta , \tan \theta , \cot \theta$.

Let $(x,y)$ be a point in X-Y plane and (r)be the length of a line segment drawn from the origin to the point.

In, $\Delta OAB,OA=r, OB=x, AB=y$

Therefore, six trigonometric functions are;

$\begin{array}{c}\sin \theta =\frac{AB}{OA}=\frac{y}{r},\csc \theta =\frac{1}{\sin \theta }=\frac{OA}{AB}=\frac{r}{y}\\ \cos \theta =\frac{OB}{OA}=\frac{x}{r},\sec \theta =\frac{1}{\cos \theta }=\frac{OA}{OB}=\frac{r}{x}\\ \tan \theta =\frac{AB}{OB}=\frac{y}{x},\cot \theta =\frac{1}{\tan \theta }=\frac{OB}{AB}=\frac{x}{y}\end{array}$

Step 3. Calculation.

The equation that can be used to find $\left(x\right)$ is written below.

$\sin x{}^{\circ }=\frac{16}{22}$

Now, the equation $\sin x{}^{\circ }=\frac{16}{22}$ is solved as,

$\begin{array}{c}\sin x°=\frac{16}{22}\\ x°={\sin }^{-1}\left(\frac{16}{22}\right)\\ x°=47°\\ x=47\end{array}$

Step 4. Conclusion.

The equation is $\sin x{}^{\circ }=\frac{16}{22}$. And the simplified value of $x=47$.