Question

find the lengths of the missing sides if side ( a ) is opposite angle ( a ), side ( b ) is opposite angle ( b ), and side ( c ) is the hypotenuse.
( a = 4, angle a = 60^{circ} )
( b=)
( c=)
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Explanation:

Step1: Find angle \( B \)

In a right - triangle, \( A + B+90^{\circ}=180^{\circ}\). Given \( A = 60^{\circ}\), then \( B=180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}\)

Step2: Use trigonometric ratios to find \( c \)

We know that \(\sin A=\frac{a}{c}\). Since \( a = 4\) and \( A = 60^{\circ}\), and \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\sin60^{\circ}=\frac{a}{c}\), so \( c=\frac{a}{\sin A}=\frac{4}{\sin60^{\circ}}=\frac{4}{\frac{\sqrt{3}}{2}}=\frac{8\sqrt{3}}{3}\)

Step3: Use trigonometric ratios to find \( b \)

We know that \(\tan A=\frac{a}{b}\). Since \( A = 60^{\circ}\) and \( a = 4\), and \(\tan60^{\circ}=\sqrt{3}\), we have \(b=\frac{a}{\tan A}=\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3}\)

Answer:

\(b = \frac{4\sqrt{3}}{3}\), \(c=\frac{8\sqrt{3}}{3}\)