# 20.2 The Sum and Difference Formulas

**Formulas for $\mathrm{sin}(\text{}\alpha \hspace{0.17em}+\hspace{0.17em}\beta )$ and $\mathrm{cos}(\text{}\alpha \hspace{0.17em}+\hspace{0.17em}\beta )$ • Formulas for $\mathrm{sin}(\text{}\alpha \hspace{0.17em}-\hspace{0.17em}\beta )$ and $\mathrm{cos}(\text{}\alpha \hspace{0.17em}-\hspace{0.17em}\beta )$ • Formulas for $\mathrm{tan}(\text{}\alpha \hspace{0.17em}+\hspace{0.17em}\beta )$ and $\mathrm{tan}(\text{}\alpha \hspace{0.17em}-\hspace{0.17em}\beta )$**

There are other important relations among the trigonometric functions. The most important and useful relations are those that involve twice an angle and half an angle. To obtain these relations, in this section, we derive the expressions for the sine and cosine of the sum and difference of two angles. These expressions will lead directly to the desired relations of double and half angles that we will derive in the following sections.

Equation (12.13), shown in the margin, gives the polar (or trigonometric) form of the product of two complex numbers. We can use this formula to derive ...

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