Question

question solve for k and w.

Explanation:

Step1: Use sine - cosine definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given $\theta = 30^{\circ}$ and hypotenuse $c = 5\sqrt{5}$.
For side $w$ (opposite to $30^{\circ}$ angle), $\sin30^{\circ}=\frac{w}{5\sqrt{5}}$. Since $\sin30^{\circ}=\frac{1}{2}$, we have $\frac{1}{2}=\frac{w}{5\sqrt{5}}$.

Step2: Solve for $w$

Cross - multiply: $w=\frac{5\sqrt{5}}{2}$.

Step3: Use cosine to find $k$

For side $k$ (adjacent to $30^{\circ}$ angle), $\cos30^{\circ}=\frac{k}{5\sqrt{5}}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{k}{5\sqrt{5}}$.

Step4: Solve for $k$

Cross - multiply: $k=\frac{5\sqrt{15}}{2}$.

Answer:

$w = \frac{5\sqrt{5}}{2}$, $k=\frac{5\sqrt{15}}{2}$