Question

b.
21°
5
x
c.
14
4
x°

Explanation:

1. For part b:

• We know that in a right - triangle, we can use the tangent function. The tangent of an angle in a right - triangle is defined as the ratio of the opposite side to the adjacent side.
• # Explanation:
• ## Step1: Identify the tangent formula

In a right - triangle with an angle $\theta$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 21^{\circ}$, the opposite side to the $21^{\circ}$ angle is $5$ and the adjacent side is $x$. So, $\tan(21^{\circ})=\frac{5}{x}$.

• ## Step2: Solve for $x$

We can re - arrange the equation $\tan(21^{\circ})=\frac{5}{x}$ to $x=\frac{5}{\tan(21^{\circ})}$. Since $\tan(21^{\circ})\approx0.3839$, then $x=\frac{5}{0.3839}\approx13.02$.

1. For part c:

• We use the tangent function again. The tangent of the angle $x^{\circ}$ in the right - triangle is the ratio of the opposite side to the adjacent side.
• # Explanation:
• ## Step1: Identify the tangent formula

$\tan x^{\circ}=\frac{\text{opposite}}{\text{adjacent}}$. Here, the opposite side to the angle $x^{\circ}$ is $4$ and the adjacent side is $14$. So, $\tan x^{\circ}=\frac{4}{14}=\frac{2}{7}\approx0.2857$.

• ## Step2: Find the angle $x$

To find $x$, we take the inverse tangent (arctan) of $\frac{2}{7}$. So, $x = \arctan(\frac{2}{7})$. Using a calculator, $x=\arctan(0.2857)\approx15.95^{\circ}$.

Answer:

b. $x\approx13.02$
c. $x\approx15.95^{\circ}$