Q24. - 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 $\text{tan17.5}{}^{\circ }=\frac{x}{23.7}$. And the simplified value of $x=7.5$.

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.

$\text{tan17.5}{}^{\circ }=\frac{x}{23.7}$

Now, the equation $\text{tan17.5}{}^{\circ }=\frac{x}{23.7}$ is solved as,

 $\begin{array}{c}\text{tan17.5}°=\frac{x}{23.7}\\ x=23.7\text{tan}\left(\text{17.5}°\right)\\ x=7.5\end{array}$

Step 4. Conclusion.

The equation is $\text{tan17.5}{}^{\circ }=\frac{x}{23.7}$. And the simplified value of $x=7.5$.