9.2 Sum and difference identities  (Page 2/6)

Begin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.

Keep in mind that we can always check the answer using a graphing calculator in radian mode.

As $\mathrm{75°}=\mathrm{45°}+\mathrm{30°},$ we can evaluate $\cos (\mathrm{75°})$ as $\cos (\mathrm{45°}+\mathrm{30°}).$

Keep in mind that we can always check the answer using a graphing calculator in degree mode.

1. Let’s begin by writing the formula and substitute the given angles.
   $$\begin{array}{ccc}\hfill \sin (\alpha -\beta )& =& \sin \alpha \cos \beta -\cos \alpha \sin \beta \hfill \\ \hfill \sin (\mathrm{45°}-\mathrm{30°})& =& \sin (\mathrm{45°})\cos (\mathrm{30°})-\cos (\mathrm{45°})\sin (\mathrm{30°})\hfill \end{array}$$

   Next, we need to find the values of the trigonometric expressions.

   $\sin (\mathrm{45°})=\frac{\sqrt{2}}{2}, \cos (\mathrm{30°})=\frac{\sqrt{3}}{2}, \cos (\mathrm{45°})=\frac{\sqrt{2}}{2}, \sin (\mathrm{30°})=\frac{1}{2}$

   Now we can substitute these values into the equation and simplify.

   $$\begin{array}{ccc}\hfill \sin (\mathrm{45°}-\mathrm{30°})& =& \frac{\sqrt{2}}{2}\left(\frac{\sqrt{3}}{2}\right)-\frac{\sqrt{2}}{2}\left(\frac{1}{2}\right)\hfill \\ & =& \frac{\sqrt{6}-\sqrt{2}}{4}\hfill \end{array}$$
2. Again, we write the formula and substitute the given angles.
   $$\begin{array}{ccc}\hfill \sin (\alpha -\beta )& =& \sin \alpha \cos \beta -\cos \alpha \sin \beta \hfill \\ \hfill \sin (\mathrm{135°}-\mathrm{120°})& =& \sin (\mathrm{135°})\cos (\mathrm{120°})-\cos (\mathrm{135°})\sin (\mathrm{120°})\hfill \end{array}$$

   Next, we find the values of the trigonometric expressions.

   $\sin (\mathrm{135°})=\frac{\sqrt{2}}{2},\cos (\mathrm{120°})=-\frac{1}{2},\cos (\mathrm{135°})=\frac{\sqrt{2}}{2},\sin (\mathrm{120°})=\frac{\sqrt{3}}{2}$

   Now we can substitute these values into the equation and simplify.

   $$\begin{array}{ccc}\hfill \sin (\mathrm{135°}-\mathrm{120°})& =& \frac{\sqrt{2}}{2}\left(-\frac{1}{2}\right)-\left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\hfill \\ & =& \frac{-\sqrt{2}+\sqrt{6}}{4}\hfill \\ & =& \frac{\sqrt{6}-\sqrt{2}}{4}\hfill \\ \sin (\mathrm{135°}-\mathrm{120°})\hfill & =& \frac{\sqrt{2}}{2}\left(-\frac{1}{2}\right)-\left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\hfill \\ & =& \frac{-\sqrt{2}+\sqrt{6}}{4}\hfill \\ & =& \frac{\sqrt{6}-\sqrt{2}}{4}\hfill \end{array}$$