Chapter 4.4, Problem 108E

(a)

To determine

To calculate:The two solutions of the equation $\tan \theta =1$ in both degrees and radians.

Answer to Problem 108E

The two solutions of the equation $\tan \theta =1$ are $\theta =45°,225°$ , and in radians $\theta =\frac{\pi }{4},\frac{5\pi }{4}$ .

Explanation of Solution

Given:

The trigonometric equation is $\tan \theta =1$ and $0°\le \theta <360°$ or $0\le \theta <2\pi$ in radians.

Calculation:

Since, the value of tangent is $1$ 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 $\tan \left(180°+\theta \right)=\tan \theta$ , therefore another value of

  $\begin{array}{c}\theta =180°+45°\\ =225°\end{array}$ .

convert $\theta =45°$ in radians.

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

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 $\tan \theta =1$ is $\theta =45°,225°$ in degrees and $\theta =\frac{\pi }{4},\frac{5\pi }{4}$ in radians.

(b)

To determine

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

Answer to Problem 108E

The two solutions of the equation $\sec \theta =\sqrt{2}$ are $\theta =30°,150°$ , and in radians $\theta =\frac{\pi }{6},\frac{5\pi }{6}$ .

Explanation of Solution

Given:

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

Calculation:

Since, the value of secant is $\sqrt{2}$ 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 $\sec \left(180°+\theta \right)=\sec \theta$ , therefore another value of

  $\begin{array}{c}\theta =180°+45°\\ =225°\end{array}$ .

Convert $\theta =45°$ in radians.

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

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 $\sec \theta =\sqrt{2}$ is $\theta =45°,225°$ in degrees and $\theta =\frac{\pi }{4},\frac{5\pi }{4}$ in radians.