Struggling . include explanation if possible

Given

two angles (150° and 12°) and the shortest side (12 cm) of a triangle

Find

the second-shortest side

Solution

Strategy: Make use of the Law of Sines to find missing side lengths when angles are known. To find the third angle, make use of the sum of angles of a triangle.

The sum of angles of a triangle is 180°, so we have

... 150° + 12° + C = 180°

... C = 180° - 162° = 18°

The law of sines tells us

... c/sin(C) = b/sin(B)

... c = sin(C)·b/sin(B) = sin(18°)·(10 cm)/sin(12°)

... c ≈ 14.9 cm

We are calling the sides a, b, c. We are calling the angles opposite those sides A, B, and C.

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