Question

what are the exact values of a and b?

a
30°
10
b
( a = 5, b = 5 sqrt { 3 } )
( a = 5, b = 10 sqrt { 3 } )
( a = 5 sqrt { 3 }, b = 5 )
( a = 5, b = 5 sqrt { 2 } )

Explanation:

Step1: Use sine function for side \(a\)

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), hypotenuse \(AB = 10\), and \(a\) is the side opposite to the \(30^{\circ}\) angle.
\(\sin30^{\circ}=\frac{a}{10}\)
Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{a}{10}\), so \(a = 10\times\frac{1}{2}=5\).

Step2: Use cosine function for side \(b\)

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), hypotenuse \(AB = 10\), and \(b\) is the side adjacent to the \(30^{\circ}\) angle.
\(\cos30^{\circ}=\frac{b}{10}\)
Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{b}{10}\), so \(b = 10\times\frac{\sqrt{3}}{2}=5\sqrt{3}\).

Answer:

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