Chapter 7.3, Problem 32AYU

To determine

To find: The trigonometric equation $\text{sin}\left(3\theta +\pi /18\right)=1$, between $0\le \theta <2\pi$.

Answer to Problem 32AYU

The solution set for the trigonometric equation $\text{sin}\left(3\theta +\pi /18\right)=1$.

$=\{-13\pi /27,\pi /6,22\pi /27\}$. $0\le \theta <2\pi$

Explanation of Solution

Given:

Trigonometric equation $\text{sin}\left(3\theta +\pi /18\right)=1$, $0\le \theta <2\pi$.

Calculation:

The angle $\theta$ for which $\text{sin}$ function has a value of $1=\pi /2$.

Therefore, the graph $y=\text{ sin}\left(3\theta +\pi /18\right)=1$ intersects the graph of $y=1$ at $\pi /2$.

But there will be other angles for which $\text{sin}\left(3\theta +\pi /18\right)=1$.

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

The period of $y=\text{ sin}\left(3\theta +\pi /18\right)=1$.

$3\theta +\pi /18=2\pi$

$3\theta =2\pi -\pi /18=35\pi /18$

The general formula of $\text{sin}\left(3\theta +\pi /18\right)=1$.

$3\theta =\pi /2+35\pi k/18$, where $k$ is any integer.

Dividing each side by 3,

$\Theta =\pi /6+35\pi k/54$, where $k$ is any integer.

For different values of $k$:

1) When $k=-1$

$\Theta =\pi /6+35\pi \left(-1\right)/54$, where $k$ is any integer.

$\Theta =\pi /6-35\pi /54$, where $k$ is any integer.

$\Theta =-26\pi /54$

2) When $k=0$

$\Theta =\pi /6+35\pi \left(0\right)/54$, where $k$ is any integer.

$\Theta =\pi /6$

3) When $k=1$

$\Theta =\pi /6+35\pi \left(1\right)/54$, where $k$ is any integer.

$\Theta =44\pi /54$

The solution set for the trigonometric equation $\text{sin}\left(3\theta +\pi /18\right)=1$ for the interval $\left[0,2\pi \right)$.

$=\{-13\pi /27,\pi /6,22\pi /27\}$. $0\le \theta <2\pi$