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.
22
20
35° a
x
a =
degrees
x =

Explanation:

Step1: Use the Law of Sines to find angle opposite to side 20

$$\frac{\sin35^{\circ}}{20}=\frac{\sin A}{22}$$
$$\sin A=\frac{22\sin35^{\circ}}{20}$$
$$\sin A=\frac{22\times0.5736}{20}$$
$$\sin A = 0.63096$$
$$A=\sin^{-1}(0.63096)\approx39.1^{\circ}$$

Step2: Use the Law of Cosines to find side \(x\)

The Law of Cosines formula is \(x^{2}=22^{2}+20^{2}-2\times22\times20\times\cos35^{\circ}\)
First, calculate \(22^{2} = 484\), \(20^{2}=400\)
\(2\times22\times20 = 880\)
\(\cos35^{\circ}\approx0.8192\)
\(x^{2}=484 + 400-880\times0.8192\)
\(x^{2}=884-720.896\)
\(x^{2}=163.104\)
\(x=\sqrt{163.104}\approx12.77\)

Answer:

\(A\approx39.1\) degrees, \(x\approx12.77\)