Chapter 7.3, Problem 29AYU

In Problems 13-36, solve each equation on the interval $0\le \theta <2\pi$.

$\sec \frac{3\theta }{2}=-2$

To determine

To find: The trigonometric equation $\text{sec}\left(3\theta /2\right)=-2$, between $0\le \theta <2\pi$.

Answer to Problem 29AYU

The solution set for the trigonometric equation $\text{sec}\left(3\theta /2\right)=-2=\left\{-8\pi /9,-4\pi /9,4\pi /9,8\pi /9,16\pi /9,-16\pi /9\right\}$ for $0\le \theta <2\pi$.

Explanation of Solution

Given:

Trigonometric equation $\text{sec}\left(3\theta /2\right)=-2$, between $0\le \theta <2\pi$.

Calculation:

Using the formula,

$\text{sec}\theta =1/\text{cos}\theta$

$\text{sec}\left(3\theta /2\right)=-2=1/\text{cos}\left(3\theta /2\right)$

$\text{cos}\left(3\theta /2\right)=-1/2$

The angle $\theta$ for which cosine function has a value of $-1/2=2\pi /3,4\pi /3$.

Therefore, the graph $y=\text{ sec}\left(3\theta /2\right)$ intersects the graph of $y=-\mathrm{½}$ at $2\pi /3,4\pi /3$.

$\text{θ}$ will be in II and III quadrants.

But there will be other angles for which $\text{cos}\left(3\theta /2\right)=-1/2$.

For the interval $\left[0,2\pi \right)$.

The period of $y=\text{ cos}\left(3\theta /2\right)$.

$3\theta /2=2\pi$

The general formula of $\text{cos}\left(3\theta /2\right)=-1/2$.

$3\theta /2=2\pi /3+2\pi k$ or $3\theta /2=4\pi /3+2\pi k$, where $k$ is any integer.

Multiplying each side by 2 and dividing each side by 3,

$\theta =4\pi /9+4\pi k/3$ or $\theta =8\pi /9+4\pi k/3$

For different values of $k$:

1. When $k=-1$.

$\theta =4\pi /9+4\pi \left(-1\right)/3$ or $\theta =8\pi /9+4\pi \left(-1\right)/3$

$\theta =4\pi /9-4\pi /3$ or $\theta =8\pi /9-4\pi /3$

$\theta =-8\pi /9$ or $\theta =-4\pi /9$

2. When $k=0$.

$\theta =4\pi /9+4\pi \left(0\right)/3$ or $\theta =8\pi /9+4\pi \left(0\right)/3$

$\theta =4\pi /9$ or $\theta =8\pi /9$

3. When $k=\text{ 1}$.

$\theta =4\pi /9+4\pi \left(1\right)/3$ or $\theta =8\pi /9+4\pi \left(1\right)/3$

$\theta =4\pi /9+4\pi /3$ or $\theta =8\pi /9+4\pi /3$

$\theta =16\pi /9$ or $\theta =20\pi /9>2\pi$

4. When $k=-2$.

$\theta =4\pi /9+4\pi \left(-2\right)/3$ or $\theta =8\pi /9+4\pi \left(-2\right)/3$

$\theta =4\pi /9-8\pi /3$ or $\theta =8\pi /9-8\pi /3$

$\theta =-20\pi /9$ outside of $2\pi$ or $\theta =-16\pi /9$

The solution set for the trigonometric equation $\text{sec}\left(3\theta /2\right)=-2$ for the interval $\left[0,2\pi \right)=\left\{-8\pi /9,-4\pi /9,4\pi /9,8\pi /9,16\pi /9,-16\pi /9\right\}$ for $0\le \theta <2\pi$.