Chapter 7.3, Problem 37AYU

In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions.

$\tan \theta =-\frac{\sqrt{3}}{3}$

To determine

To find: The trigonometric equation $\text{tan }\theta =-\sqrt{3}/3$ by writing the general formula, and to obtain 6 solutions.

Answer to Problem 37AYU

The general formula of $\text{tan }\theta =-\sqrt{3}/3$.

$\theta =5\pi /6+\pi k$ or $\theta =11\pi /6+\pi k$, where $k$ is any integer.

The solution set for the trigonometric equation $\text{tan }\theta =-\sqrt{3}/3$.

The six solutions $=\{-\pi /6,5\pi /6,-7\pi /6,-13\pi /6,11\pi /6,17\pi /6\}$.

Explanation of Solution

Given:

Trigonometric equation $\text{tan }\theta =-\sqrt{3}/3$.

Calculation:

Trigonometric equation $\text{tan }\theta =-\sqrt{3}/3<0$, therefore, $\theta$ is in II and IV quadrants.

The angle $\theta$ for which $\text{tan}$ function has a value of $-\sqrt{3}/3=5\pi /6,11\pi /6$.

Therefore, the graph $y=\text{ tan }\theta$ intersects the graph of $y=-\sqrt{3}/3$ at $5\pi /6,11\pi /6$.

The period of $y=\text{ tan }\theta$.

$\theta =\pi$

The general formula of $\text{tan }\theta =-\sqrt{3}/3$.

$\theta =5\pi /6+\pi k$ or $\theta =11\pi /6+\pi k$, where $k$ is any integer.

For different values of $k$:

1) When $k=-1$

$\theta =5\pi /6+\pi \left(-1\right)$ or $\theta =11\pi /6+\pi \left(-1\right)$, where $k$ is any integer.

$\Theta =5\pi /6-\pi$ or $\theta =11\pi /6-\pi$

$\theta =-\pi /6$ or $5\pi /6$

2) When $k=-2$

$\theta =5\pi /6+\pi \left(-2\right)$ or $\theta =11\pi /6+\pi \left(-2\right)$, where $k$ is any integer.

$\Theta =5\pi /6-\mathrm{<mn>2</mn>\pi }$ or $\theta =11\pi /6-\mathrm{<mn>2</mn>\pi }$

$\theta =-\mathrm{<mn>7</mn>\pi }/6$ or $-\pi /6$

3) When $k=-3$

$\theta =5\pi /6+\pi \left(-3\right)$ or $\theta =11\pi /6+\pi \left(-3\right)$, where $k$ is any integer.

$\Theta =5\pi /6-\mathrm{<mn>3</mn>\pi }$ or $\theta =11\pi /6-\mathrm{<mn>3</mn>\pi }$

$\Theta =-\mathrm{<mn>13</mn>\pi }/6$ or $-7\pi /6$

4) When $k=0$

$\theta =5\pi /6+\pi \left(0\right)$ or $\theta =11\pi /6+\pi \left(0\right)$, where $k$ is any integer.

$\Theta =5\pi /6$ or $11\pi /6$

5) When $k=1$

$\theta =5\pi /6+\pi \left(1\right)$ or $\theta =11\pi /6+\pi \left(1\right)$, where $k$ is any integer.

$\Theta =5\pi /6+\pi$ or $\theta =11\pi /6+\pi$

$\Theta =11\pi /6$ or $17\pi /6$

The solution set for the trigonometric equation $\text{tan }\theta =-\sqrt{3}/3$.

$=\{-\pi /6,5\pi /6,-7\pi /6,-13\pi /6,11\pi /6,17\pi /6\}$