7.2 Sum and difference identities  (Page 2/6)

Use the formula for the cosine of the difference of two angles. We have

As ${75}^{\circ }={45}^{\circ }+{30}^{\circ },$ we can evaluate $\cos \left({75}^{\circ }\right)$ as $\cos \left({45}^{\circ }+{30}^{\circ }\right).$ Thus,

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

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

   $\sin \left({45}^{\circ }\right)=\frac{\sqrt{2}}{2},\cos \left({30}^{\circ }\right)=\frac{\sqrt{3}}{2},\cos \left({45}^{\circ }\right)=\frac{\sqrt{2}}{2},\sin \left({30}^{\circ }\right)=\frac{1}{2}$

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

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

   Next, we find the values of the trigonometric expressions.

   $\sin \left({135}^{\circ }\right)=\frac{\sqrt{2}}{2},\cos \left({120}^{\circ }\right)=-\frac{1}{2},\cos \left({135}^{\circ }\right)=\frac{\sqrt{2}}{2},\sin \left({120}^{\circ }\right)=\frac{\sqrt{3}}{2}$

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

   $$\begin{array}{l}\sin ({135}^{\circ }-{120}^{\circ })=\frac{\sqrt{2}}{2}\left(-\frac{1}{2}\right)-\left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\hfill \\ =\frac{-\sqrt{2}+\sqrt{6}}{4}\hfill \\ =\frac{\sqrt{6}-\sqrt{2}}{4}\hfill \\ \hfill \\ \sin ({135}^{\circ }-{120}^{\circ })=\frac{\sqrt{2}}{2}\left(-\frac{1}{2}\right)-\left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\hfill \\ =\frac{-\sqrt{2}+\sqrt{6}}{4}\hfill \\ =\frac{\sqrt{6}-\sqrt{2}}{4}\hfill \end{array}$$