Question

what are the exact values of a and b? a = 5, b = 5√2 a = 5√3, b = 5 a = 5, b = 5√3 a = 5, b = 10√3

Explanation:

Step1: Use sine - cosine relations in right - triangle

In right - triangle \(ABC\) with \(\angle A = 30^{\circ}\) and hypotenuse \(AB = 10\). We know that \(\sin A=\frac{BC}{AB}\) and \(\cos A=\frac{AC}{AB}\).
Since \(\sin30^{\circ}=\frac{1}{2}\) and \(\sin A=\frac{a}{10}\), then \(a = 10\times\sin30^{\circ}\).
\[a = 10\times\frac{1}{2}=5\]

Step2: Use cosine function to find \(b\)

Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\) and \(\cos A=\frac{b}{10}\), then \(b = 10\times\cos30^{\circ}\).
\[b = 10\times\frac{\sqrt{3}}{2}=5\sqrt{3}\]

Answer:

\(a = 5,b = 5\sqrt{3}\)