Question

what are the exact values of a and b?
o a = 7√3/2, b = 7/2
o a = 7/2, b = 7√3
o a = 7/2, b = 7√2/2
o a = 7/2, b = 7√3/2

Explanation:

Step1: Use sine - cosine definitions in right - triangle

In right - triangle \(ABC\) with \(\angle A = 30^{\circ}\), hypotenuse \(AB = 7\), \(\sin A=\frac{a}{AB}\) and \(\cos A=\frac{b}{AB}\).

Step2: Calculate the value of \(a\)

Since \(\sin A=\sin30^{\circ}=\frac{1}{2}\) and \(\sin A=\frac{a}{AB}\), and \(AB = 7\), then \(a = AB\times\sin A=7\times\frac{1}{2}=\frac{7}{2}\).

Step3: Calculate the value of \(b\)

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

Answer:

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