Question

find the missing side lengths. leave your answers in simplified radical form.
4√3
30°
x
x = 4
y = 4
x = 8
y = 4√3/3
x = 8
y = 4
x = 4
y = 4√3/3

Explanation:

Step1: Use sine - cosine relations

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given $\theta = 30^{\circ}$ and the hypotenuse is $x$, and the side opposite to the $30^{\circ}$ angle is $y$, and the side adjacent to the $30^{\circ}$ angle is $4\sqrt{3}$. We know that $\cos30^{\circ}=\frac{4\sqrt{3}}{x}$.
Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have the equation $\frac{\sqrt{3}}{2}=\frac{4\sqrt{3}}{x}$.
Cross - multiply: $\sqrt{3}x = 8\sqrt{3}$.
Solve for $x$: $x = 8$.

Step2: Use sine relation

We know that $\sin30^{\circ}=\frac{y}{x}$. Since $x = 8$ and $\sin30^{\circ}=\frac{1}{2}$, then $\frac{1}{2}=\frac{y}{8}$.
Solve for $y$: $y = 4$.

Answer:

$x = 8$, $y = 4$