## Precalculus (6th Edition) Blitzer

Let us consider the left side of the given expression: $\sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)$ Now, evaluate the expression and substitute the appropriate values: \begin{align} & \sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)=\sin {{90}^{{}^\circ }} \\ & =1 \end{align} And consider the right side of the provided expression: $\sin {{30}^{{}^\circ }}+\sin {{60}^{{}^\circ }}$ Now, evaluate the expression and substitute the appropriate values: \begin{align} & \sin {{30}^{{}^\circ }}+\sin {{60}^{{}^\circ }}=\frac{1}{2}+\frac{\sqrt{3}}{2} \\ & =\frac{1+\sqrt{3}}{2} \end{align} Thus, the left side of the given expression is not equal to the right side, which is $\sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)$ or $\sin {{90}^{{}^\circ }}$ and equal to $\sin {{30}^{{}^\circ }}+\sin {{60}^{{}^\circ }}$. Hence, they are not equal. (b) Let us consider the left side of the given expression: $\sin \left( 30{}^\circ +60{}^\circ \right)$ Now, evaluate the expression and substitute the appropriate values: \begin{align} & \sin \left( {{30}^{{}^\circ }}+{{60}^{{}^\circ }} \right)=\sin {{90}^{{}^\circ }} \\ & =1 \end{align} Now, consider the right side of the provided expression: $\sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ$ And, evaluate the expression and substitute the appropriate values: \begin{align} & \sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ ~=\left( \frac{1}{2} \right)\left( \frac{1}{2} \right)+\left( \frac{\sqrt{3}}{2} \right)\left( \frac{\sqrt{3}}{2} \right) \\ & =\frac{1}{4}+\frac{3}{4} \\ & =1 \end{align} Thus, the left side of the given expression is equal to the right side, which is $\sin \left( 30{}^\circ +60{}^\circ \right)$ or $\sin 90{}^\circ$ and equal to $\sin 30{}^\circ \cos 60{}^\circ +\cos 30{}^\circ \sin 60{}^\circ$. Hence, they are equal.