Question

a) u = 6, v = 3
b) u = 6, v = \frac{3}{2}
c) u = 3, v = 3
d) u = 3, v = \frac{3}{2}

Explanation:

Step1: Use cosine function

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), adjacent side to \(30^{\circ}\) is \(3\sqrt{3}\), and hypotenuse is \(u\). So \(\cos30^{\circ}=\frac{3\sqrt{3}}{u}\). Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{u}\). Cross - multiply: \(\sqrt{3}u = 6\sqrt{3}\), then \(u = 6\).

Step2: Use sine function

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), opposite side to \(30^{\circ}\) is \(v\), and hypotenuse is \(u = 6\). So \(\sin30^{\circ}=\frac{v}{u}\). Since \(\sin30^{\circ}=\frac{1}{2}\) and \(u = 6\), then \(v=\frac{1}{2}\times u\). Substitute \(u = 6\) into the formula, we get \(v = 3\).

Answer:

A. \(u = 6,v = 3\)