Question

given the triangle below, find the angle a and length of side x
note: picture is not drawn to scale, but you can assume an angle that appears acute is acute and angle that appears obtuse is obtuse.
(image of triangle with sides 32, 24, x and angles 32°, a)
a = degrees
x =

Explanation:

Step1: Use the Law of Sines to find angle opposite to side 24

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Let the angle opposite to side 24 be \(B\). Then \(\frac{\sin32^{\circ}}{24}=\frac{\sin B}{32}\). Cross - multiply: \(32\sin32^{\circ}=24\sin B\). So \(\sin B=\frac{32\sin32^{\circ}}{24}\). Calculate \(\sin32^{\circ}\approx0.5299\), then \(\sin B=\frac{32\times0.5299}{24}\approx0.7065\). Since the angle appears acute (from the note), \(B = \sin^{- 1}(0.7065)\approx45^{\circ}\)

Step2: Find angle \(A\)

Using the angle - sum property of a triangle (\(A + B+32^{\circ}=180^{\circ}\)), so \(A=180^{\circ}-32^{\circ}-45^{\circ}=103^{\circ}\)

Step3: Use the Law of Sines to find \(x\)

\(\frac{\sin32^{\circ}}{24}=\frac{\sin103^{\circ}}{x}\). Cross - multiply: \(x\sin32^{\circ}=24\sin103^{\circ}\). \(\sin103^{\circ}\approx0.9744\), \(\sin32^{\circ}\approx0.5299\). Then \(x=\frac{24\times0.9744}{0.5299}\approx44\)

Answer:

\(A = 103\) degrees, \(x = 44\)