Find the values of x and y.

Steps to Solve

1. Understanding the triangle components

In a right triangle with a 30-degree angle, we can identify the sides relative to the angle:

• The side opposite the 30-degree angle is 9 (this is the side labeled as "y").
• The hypotenuse is the longest side, which we will label as "h".
• The adjacent side (labeled as "x") is the side adjacent to the 30-degree angle.

2. Using the sine function

To find the length of the hypotenuse "h", we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse.

Using the sine function:

$$ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{h} $$

Substituting the known values:

$$ \sin(30^\circ) = \frac{9}{h} $$

Since $\sin(30^\circ) = \frac{1}{2}$, we can set up the equation:

$$ \frac{1}{2} = \frac{9}{h} $$

3. Solving for the hypotenuse "h"

Rearranging the equation to solve for "h":

$$ h = 9 \cdot 2 = 18 $$

So, the hypotenuse "h" is 18.

4. Using the cosine function to find "x"

Now we can find the adjacent side "x" using the cosine function. The cosine of the angle is given by the ratio of the adjacent side to the hypotenuse:

Using the cosine function:

$$ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{h} $$

Substituting the known values:

$$ \cos(30^\circ) = \frac{x}{18} $$

Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, we can set up the equation:

$$ \frac{\sqrt{3}}{2} = \frac{x}{18} $$

5. Solving for "x"

Rearranging to find "x":

$$ x = 18 \cdot \frac{\sqrt{3}}{2} = 9\sqrt{3} $$