Algebra and Trigonometry, 5th Edition

8.1 The Law of Sines

• Use the law of sines to solve triangles.

• Find the area of any triangle given the lengths of two sides and the measure of the included angle.

To solve a triangle means to find the lengths of all its sides and the measures of all its angles. We solved right triangles in Section 6.2. For review, let’s solve the right triangle shown below. We begin by listing the known measures:

$$\begin{array}{ccc}Q\hfill & =\hfill & 37.1°,\hfill \\ W\hfill & =\hfill & 90°\hfill \\ Z\hfill & =\hfill & ?,\hfill \end{array}\begin{array}{ccc}q\hfill & =\hfill & ?,\hfill \\ w\hfill & =\hfill & ?,\hfill \\ z\hfill & =\hfill & \mathrm{6.3.}\hfill \end{array}$$

Since the sum of the three angle measures of any triangle is $180°$, we can immediately find the measure of the third angle:

$$\begin{array}{ccc}Z\hfill & \hfill =\hfill & 180°-\left(90°+37.1°\right)\hfill \\ \hfill =\hfill & 52.9°\hfill \end{array}$$

Then using the tangent ratio and the cosine ratio, respectively, ...