Q23. - Algebra 2

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.

Step-by-Step Solution

Answer

The equation is $\csc 54{}^{\circ }=\frac{x}{17.8}$. And the simplified value of $x=22$.

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.

$\csc 54{}^{\circ }=\frac{x}{17.8}$

Now, the equation $\csc 54{}^{\circ }=\frac{x}{17.8}$ is solved as,

$\begin{array}{c}\csc 54°=\frac{x}{17.8}\\ x=17.8\csc \left(54°\right)\\ x=22\end{array}$ 

Step 4. Conclusion.

The equation is $\csc 54{}^{\circ }=\frac{x}{17.8}$. And the simplified value of $x=22$.