Question

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

Short answer

The equation is$\text{cos23}{}^{\circ }=\frac{32}{x}$.And the simplified value of$x=34.8$.

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.

$\text{cos23}{}^{\circ }=\frac{32}{x}$

Now, the equation $\text{cos23}{}^{\circ }=\frac{32}{x}$ is solved as,

$\begin{array}{c}\text{cos23}°=\frac{32}{x}\\ x=\frac{32}{\cos 23°}\\ x=32\sec \left(23°\right)\\ x=34.8\end{array}$

Step 4. Conclusion.

The equation is $\text{cos23}{}^{\circ }=\frac{32}{x}$. And the simplified value of $x=34.8$.