Chapter 4.4, Problem 104E

(a)

To determine

To calculate:The two solutions of the equation $\cos \theta =\frac{\sqrt{2}}{2}$ in both degrees and radians.

Answer to Problem 104E

The two solutions of the equation $\cos \theta =\frac{\sqrt{2}}{2}$ are $\theta =45°,315°$ , and in radians $\theta =\frac{\pi }{4},\frac{7\pi }{4}$ .

Explanation of Solution

Given Information:

The trigonometric equation is $\cos \theta =\frac{\sqrt{2}}{2}$ and $0°\le \theta <360°$ or $0\le \theta <2\pi$ in radians.

Calculation:

Since, the value of cos is $\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$ and it gives the internal angles are $45°,90°$ and $45°$ . The opposite side is $1$ and the hypotenuse is $\sqrt{2}$ . The value of base is $1$ . Draw a triangle

  

So, the angle $\theta =45°$ and as per the trigonometric identity $\cos \left(360°-\theta \right)=\cos \theta$ , therefore another value of

  $\begin{array}{c}\theta =360°-45°\\ =315°\end{array}$ .

Convert $\theta =45°$ in radians.

  $\begin{array}{c}\theta =45°\times \frac{\pi }{180°}\\ =\frac{\pi }{4}\end{array}$

For $\theta =315°$ in radians

  $\begin{array}{c}\theta =315°\times \frac{\pi }{180°}\\ =\frac{7\pi }{4}\end{array}$ .

Therefore the values of $\theta$ for $\cos \theta =\frac{\sqrt{2}}{2}$ is $\theta =45°,315°$ in degrees and $\theta =\frac{\pi }{4},\frac{7\pi }{4}$ in radians.

(b)

To determine

To calculate:The two solutions of the equation $\cos \theta =-\frac{\sqrt{2}}{2}$ in both degrees and radians.

Answer to Problem 104E

The two solutions of the equation $\cos \theta =-\frac{\sqrt{2}}{2}$ are $\theta =135°,225°$ , and in radians $\theta =\frac{3\pi }{4},\frac{5\pi }{4}$ .

Explanation of Solution

Given Information:

The trigonometric equation is $\cos \theta =-\frac{\sqrt{2}}{2}$ and $0°\le \theta <360°$ or $0\le \theta <2\pi$ in radians.

Calculation:

Since, the value of cos is $-\frac{\sqrt{2}}{2}$ and by the trigonometric property $\cos \left(180°-\theta \right)=-\cos \theta$ and as $-\cos 45°=-\frac{\sqrt{2}}{2}$ .

So, the angle

  $\begin{array}{c}\theta =180°-45°\\ =135°\end{array}$

As per the trigonometric identity, $\cos \left(360°-\theta \right)=\cos \theta$ , So $\cos \left(360°-\theta \right)=\cos 135°$ , therefore another value of

  $\begin{array}{c}\theta =360°-135°\\ =225°\end{array}$

Convert $\theta =135°$ in radians0

  $\begin{array}{c}\theta =135°\times \frac{\pi }{180°}\\ =\frac{3\pi }{4}\end{array}$

For $\theta =225°$ in radians

  $\begin{array}{c}\theta =225°\times \frac{\pi }{180°}\\ =\frac{5\pi }{4}\end{array}$ .

Therefore the values of $\theta$ for $\cos \theta =-\frac{\sqrt{2}}{2}$ is $\theta =135°,225°$ in degrees and $\theta =\frac{3\pi }{4},\frac{5\pi }{4}$ in radians.