Chapter 13.6, Problem 31PPS

To determine

To find the exact value of $\cos {45}^{\circ }-\cos {30}^{\circ }$ .

Answer to Problem 31PPS

  $\frac{1-\sqrt{3}}{2}$

Explanation of Solution

Given:

  $\cos {45}^{\circ }-\cos {30}^{\circ }$

Calculation:

Given expression $\cos {45}^{\circ }-\cos {30}^{\circ },$

  $\cos {45}^{\circ }-\cos {30}^{\circ }=\frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{2}$   [ $\because \cos {45}^{\circ }=\frac{1}{\sqrt{2}},\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$ ]

  $=\frac{1\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}-\frac{\sqrt{3}}{2}$   [Rationalizing first ratio]

  $\begin{array}{l}=\frac{1}{2}-\frac{\sqrt{3}}{2}\\ =\frac{1-\sqrt{3}}{2}\end{array}$

Conclusion:

Therefore, the exact value of $\cos {45}^{\circ }-\cos {30}^{\circ }$ is $\frac{1-\sqrt{3}}{2}$ .