Question

what are the exact values of a and b?

a
30°
7
b
c
b
a

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

Explanation:

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

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Given \(\theta = 30^{\circ}\) and hypotenuse \(AB = 7\), for side \(a\) (opposite to \(30^{\circ}\) angle), \(\sin30^{\circ}=\frac{a}{7}\). Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(a = 7\times\sin30^{\circ}=7\times\frac{1}{2}=\frac{7}{2}\).

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

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Given \(\theta = 30^{\circ}\) and hypotenuse \(AB = 7\), for side \(b\) (adjacent to \(30^{\circ}\) angle), \(\cos30^{\circ}=\frac{b}{7}\). Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(b = 7\times\cos30^{\circ}=7\times\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{2}\).

Answer:

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