Question

what are the exact values of a and b?

Explanation:

Step1: Use trigonometric ratios

In a right - angled triangle with an angle of \(30^{\circ}\), \(\sin30^{\circ}=\frac{a}{7}\) and \(\cos30^{\circ}=\frac{b}{7}\).
Since \(\sin30^{\circ}=\frac{1}{2}\), then \(a = 7\times\sin30^{\circ}\).

$$a = 7\times\frac{1}{2}=\frac{7}{2}$$

Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), then \(b = 7\times\cos30^{\circ}\).

$$b = 7\times\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{2}$$

Answer:

\(a=\frac{7}{2},b = \frac{7\sqrt{3}}{2}\) (the second option)