Question

the law of sines: the ambiguous case
solve 88°, a = 110 ft m, c = 54 ft
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.
(triangle with sides 32, 26, x and angle 29° at the left, angle a at the bottom right)
a = degrees
x =

Explanation:

Step1: Use the Law of Sines to find angle opposite to side 26

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Let's assume the side of length 32 is opposite the \(29^{\circ}\) angle. Let the side of length 26 be opposite angle \(B\). Then \(\frac{\sin29^{\circ}}{32}=\frac{\sin B}{26}\). Cross - multiply: \(26\sin29^{\circ}=32\sin B\). So, \(\sin B=\frac{26\sin29^{\circ}}{32}\). Calculate \(\sin29^{\circ}\approx0.4848\). Then \(\sin B=\frac{26\times0.4848}{32}\approx0.394\). Using the inverse - sine function, \(B = \sin^{-1}(0.394)\approx23.2^{\circ}\).

Step2: Calculate angle \(A\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(A=180^{\circ}-29^{\circ}-B\). Substitute \(B\approx23.2^{\circ}\), then \(A = 180^{\circ}-29^{\circ}-23.2^{\circ}=127.8^{\circ}\).

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

Using the Law of Sines \(\frac{\sin A}{x}=\frac{\sin29^{\circ}}{32}\). Substitute \(A = 127.8^{\circ}\) and \(\sin127.8^{\circ}=\sin(180 - 52.2)^{\circ}=\sin52.2^{\circ}\approx0.790\). Then \(x=\frac{32\sin127.8^{\circ}}{\sin29^{\circ}}\). Since \(\sin29^{\circ}\approx0.4848\), \(x=\frac{32\times0.790}{0.4848}\approx52.1\).

Answer:

\(A\approx127.8\) degrees, \(x\approx52.1\)