Question

find the length of side $x$ in simplest radical form with a rational denominator.
triangle image: right triangle with angles 30°, 60°, 90°, hypotenuse 11, side $x$ opposite 30°? wait, no, angle at top left 60°, bottom 30°, right angle, hypotenuse 11, side $x$ is one leg.

Explanation:

Step1: Identify trigonometric ratio

We use cosine for the 60° angle: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta=60^\circ$, adjacent side is $x$, hypotenuse is 11.

Step2: Substitute values into formula

$\cos(60^\circ) = \frac{x}{11}$
We know $\cos(60^\circ) = \frac{1}{2}$, so:
$\frac{1}{2} = \frac{x}{11}$

Step3: Solve for $x$

Multiply both sides by 11:
$x = 11 \times \frac{1}{2} = \frac{11}{2}$

Answer:

$\frac{11}{2}$