Chapter 6, Problem 10RE

To determine

Find the exact value of the given expression.

Answer to Problem 10RE

The value of the given expression is $0$ .

Explanation of Solution

Given:

The given expression is $\cos {540}^{°}-\tan \left(-{405}^{°}\right)$ radians.

Calculation:

  $\begin{array}{l}\cos {540}^{°}-\tan \left(-{405}^{°}\right)\\ =\cos \left({360}^{°}+{180}^{°}\right)-\tan \left(-{360}^{°}-{45}^{°}\right)\\ =\cos {540}^{°}-\frac{\sin \left(-{360}^{°}-{45}^{°}\right)}{\cos \left(-{360}^{°}-{45}^{°}\right)}\end{array}$

Draw a unit circle diagram for trigonometric values.

  

Take clockwise as negative angle direction and counterclockwise as positive angle direction.

From diagram,

  $\cos {180}^{°}=-1,\sin {315}^{°}=-\frac{\sqrt{2}}{2},\cos {315}^{°}=\frac{\sqrt{2}}{2}$

  $\begin{array}{l}=\left(-1\right)-\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\\ =-1-\left(-\frac{\sqrt{2}}{2}\right)\cdot \frac{2}{\sqrt{2}}\\ =-1+1\\ =0\end{array}$

Hence thevalue of the given expressionis $0$ .