Find the values of x and y.

Steps to Solve

1. Identify the triangle properties

The triangle is a right triangle with one angle measuring $30^\circ$. The side opposite the $30^\circ$ angle is given as $9$.

2. Using the sine function for side ( y )

In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. We can use: $$ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} $$ Thus, $$ \sin(30^\circ) = \frac{9}{y} $$

Since ( \sin(30^\circ) = \frac{1}{2} ), we have: $$ \frac{1}{2} = \frac{9}{y} $$

3. Solve for ( y )

Multiply both sides by ( y ) and then by ( 2 ): $$ y = 18 $$

4. Using the cosine function for side ( x )

Now we can use the cosine function for the angle to find side ( x ): $$ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} $$ This translates to: $$ \cos(30^\circ) = \frac{x}{9} $$

5. Solve for ( x )

Since ( \cos(30^\circ) = \frac{\sqrt{3}}{2} ), we have: $$ \frac{\sqrt{3}}{2} = \frac{x}{9} $$

Multiply both sides by ( 9 ): $$ x = 9 \cdot \frac{\sqrt{3}}{2} $$ Thus, $$ x = \frac{9\sqrt{3}}{2} $$