Example 5:

Find θ to the nearest degree in this triangle:

Solution:

Again we have an SSA case, but we cannot find θ directly since the side opposite it is also unknown.  Therefore we will find the angle opposite the side of length 20 and then use the fact that the angles in a triangle add to 180^o:

Since the side opposite the 40^o angle is greater than the adjacent side, there can only be one such triangle, and α is less than 90^o so the Law of Sines gives its value directly:

Therefore θ = 180^o – 40^o – 31^o = 109^o

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