Question

for problems #6 - 7, solve for x using right - triangle trigonometry. write an equation and show your work. write your final answer in the blanks rounded to the nearest hundredth.

1. x=
2. x=

Explanation:

Step1: Identify trig - ratio for problem 6

In the first right - triangle, we know the adjacent side to the $31^{\circ}$ angle is $x$ and the opposite side is $13$. We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. So, $\tan31^{\circ}=\frac{13}{x}$.
$$x = \frac{13}{\tan31^{\circ}}$$

Step2: Calculate the value of $x$ for problem 6

We know that $\tan31^{\circ}\approx0.6009$. Then $x=\frac{13}{0.6009}\approx21.64$.

Step3: Identify trig - ratio for problem 7

In the second right - triangle, we know the hypotenuse is $5$ and the adjacent side to the $31^{\circ}$ angle is $x$. We use the cosine function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So, $\cos31^{\circ}=\frac{x}{5}$.
$$x = 5\times\cos31^{\circ}$$

Step4: Calculate the value of $x$ for problem 7

We know that $\cos31^{\circ}\approx0.8572$. Then $x = 5\times0.8572=4.29$.

Answer:

1. $x\approx21.64$
2. $x\approx4.29$