Question

question solve for w and k. answer attempt 1 out of 4 k = w =

Explanation:

Step1: Use sine - cosine relationships

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given an angle of $30^{\circ}$, and the side adjacent to the $30^{\circ}$ angle is $3$.
We know that $\cos30^{\circ}=\frac{3}{w}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{3}{w}$.

Step2: Solve for $w$

Cross - multiply: $\sqrt{3}w = 6$, then $w=\frac{6}{\sqrt{3}}$. Rationalize the denominator: $w = 2\sqrt{3}$.

Step3: Use tangent relationship to find $k$

We know that $\tan30^{\circ}=\frac{k}{3}$. Since $\tan30^{\circ}=\frac{1}{\sqrt{3}}$, we have $\frac{1}{\sqrt{3}}=\frac{k}{3}$.

Step4: Solve for $k$

Cross - multiply: $\sqrt{3}k=3$, then $k = \sqrt{3}$.

Answer:

$k=\sqrt{3}$
$w = 2\sqrt{3}$