Basic Technical Mathematics, 11th Edition

9.6 The Law of Cosines

• Law of Cosines • Case 3: Two Sides & Included Angle • Case 4: Three Sides • Summary of Solving Oblique Triangles

As noted in the last section, the law of sines cannot be used for Case 3 (two sides and the included angle) and Case 4 (three sides). In this section, we develop the law of cosines, which can be used for Cases 3 and 4. After finding another part of the triangle using the law of cosines, we will often find it easier to complete the solution using the law of sines.

Consider any oblique triangle—for example, either triangle shown in Fig. 9.68. For each triangle, $h\hspace{0.17em}/\hspace{0.17em}b\hspace{0.17em}=\hspace{0.17em}\sin A\hspace{0.17em},\hspace{0.17em}$ or $h\hspace{0.17em}=\hspace{0.17em}b\sin A\hspace{0.17em}.\hspace{0.17em}$ Also, using the Pythagorean theorem, we obtain $a^2\hspace{0.17em}=\hspace{0.17em}{h}^2\hspace{0.17em}+\hspace{0.17em}{x}^2$ for each triangle. Therefore [with ${(\sin A)}^2\hspace{0.17em}=\hspace{0.17em}{\sin }^{2}A$], ...