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# Chapter 7: Applications of Trigonometric Functions ## 7.1 Oblique Triangles ### The Law of Sines #### Derivation of the Law of Sines Consider triangle $ABC$. $h = b\sin{A}$ and $h = a\sin{B}$ Therefore, $$ \begin{aligned} a\sin{B} &= b\sin{A} \\ \frac{a}{\sin{A}}&=\frac{b}{\sin{B}} \end{alig...

# Chapter 7: Applications of Trigonometric Functions ## 7.1 Oblique Triangles ### The Law of Sines #### Derivation of the Law of Sines Consider triangle $ABC$. $h = b\sin{A}$ and $h = a\sin{B}$ Therefore, $$ \begin{aligned} a\sin{B} &= b\sin{A} \\ \frac{a}{\sin{A}}&=\frac{b}{\sin{B}} \end{aligned} $$ Similarly, it can be shown that $\frac{a}{\sin{A}} = \frac{c}{\sin{C}}$. **Law of Sines** In triangle $ABC$, $\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}$. **Example 1** Solve triangle $ABC$ if $A = 32^\circ$, $B = 81.8^\circ$, and $a = 42.9$ cm. **Solution** $C = 180^\circ - A - B = 180^\circ - 32^\circ - 81.8^\circ = 66.2^\circ$ $\frac{a}{\sin{A}} = \frac{b}{\sin{B}}$ $b = \frac{a\sin{B}}{\sin{A}} = \frac{42.9\sin{81.8^\circ}}{\sin{32^\circ}} \approx 80.1$ cm $\frac{a}{\sin{A}} = \frac{c}{\sin{C}}$ $c = \frac{a\sin{C}}{\sin{A}} = \frac{42.9\sin{66.2^\circ}}{\sin{32^\circ}} \approx 74.1$ cm **Answer**: $C= 66.2^\circ$, $b = 80.1$ cm, and $c = 74.1$ cm