Question

what are the exact values of a and b?
a
b
c
30°
10
b
o a = 5, b = 10√3
o a = 5√3, b = 5
o a = 5, b = 5√2
o a = 5, b = 5√3

Explanation:

Step1: Use sine function for side a

In right - triangle \(ABC\), \(\sin A=\frac{a}{AB}\). Given \(A = 30^{\circ}\) and \(AB = 10\). Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(a=AB\times\sin A\). So \(a = 10\times\sin30^{\circ}=10\times\frac{1}{2}=5\).

Step2: Use cosine function for side b

\(\cos A=\frac{b}{AB}\). Given \(A = 30^{\circ}\) and \(AB = 10\), and \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\). Then \(b = AB\times\cos A=10\times\frac{\sqrt{3}}{2}=5\sqrt{3}\).

Answer:

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