Question

in the triangle, angle b measures 60° and bc is 18. what is the length of ac?

Explanation:

Step1: Recall trigonometric ratio

In right - triangle ABC with right - angle at A, $\cos B=\frac{AB}{BC}$ and $\sin B=\frac{AC}{BC}$. Given $\angle B = 60^{\circ}$ and $BC = 18$. We use the sine function since we want to find the length of the side opposite to $\angle B$.

Step2: Apply sine formula

The sine of an angle in a right - triangle is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\sin B=\sin60^{\circ}=\frac{AC}{BC}$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$ and $BC = 18$, we have $\frac{\sqrt{3}}{2}=\frac{AC}{18}$.

Step3: Solve for AC

Cross - multiply the equation $\frac{\sqrt{3}}{2}=\frac{AC}{18}$ to get $AC = 18\times\frac{\sqrt{3}}{2}$. Simplifying, $AC = 9\sqrt{3}$.

Answer:

$9\sqrt{3}$