Chapter 7.3, Problem 36AYU

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

$\tan \theta =1$

To determine

To find: The trigonometric equation $\text{tan }\theta =1$ by writing the general formula, and to obtain 6 solutions.

Answer to Problem 36AYU

The general formula of $\text{tan }\theta =1$.

$\theta =\pi /4+\pi k$ or $\theta =5\pi /4+\pi k$, where $k$ is any integer.

The solution set for the trigonometric equation $\text{tan }\theta =1$.

The six solutions $=\{-3\pi /4,\pi /4,-7\pi /4,-11\pi /4,5\pi /4,9\pi /4\}$.

Explanation of Solution

Given:

Trigonometric equation $\text{tan }\theta =1$.

Calculation:

Trigonometric equation $\text{tan }\theta =1>0$, therefore, $\theta$ is in I and III quadrants.

The angle $\theta$ for which $\text{tan}$ function has a value of $1=\pi /4,5\pi /4$.

Therefore, the graph $y=\text{ tan }\theta$ intersects the graph of $y=1$ at $\pi /4,5\pi /4$.

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

$\theta =\pi$

The general formula of $\text{tan }\theta =1$.

$\theta =\pi /4+\pi k$ or $\theta =5\pi /4+\pi k$, where $k$ is any integer.

For different values of $k$:

1) When $k=-1$

$\theta =\pi /4+\pi \left(-1\right)$ or $\theta =5\pi /4+\pi \left(-1\right)$, where $k$ is any integer.

$\Theta =\pi /4-\pi$ or $\theta =5\pi /4-\pi$

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

2) When $k=-2$

$\theta =\pi /4+\pi \left(-2\right)$ or $\theta =5\pi /4+\pi \left(-2\right)$, where $k$ is any integer.

$\Theta =\pi /4-\text{ 2}\pi$ or $\theta =5\pi /4-\text{ 2}\pi$

$\theta =-7\pi /4$ or $-3\pi /4$

3) When $k=-3$

$\theta =\pi /4+\pi \left(-3\right)$ or $\theta =5\pi /4+\pi \left(-3\right)$, where $k$ is any integer.

$\Theta =\pi /4-\text{ 3}\pi$ or $\theta =5\pi /4-\text{ 3}\pi$

$\Theta =-11\pi /4$ or $-7\pi /4$

4) When $k=0$

$\theta =\pi /4+\pi \left(0\right)$ or $\theta =5\pi /4+\pi \left(0\right)$, where $k$ is any integer.

$\Theta =\pi /4$ or $5\pi /4$

5) When $k=1$

$\theta =\pi /4+\pi \left(1\right)$ or $\theta =5\pi /4+\pi \left(1\right)$, where $k$ is any integer.

$\Theta =\pi /4+\pi$ or $\theta =5\pi /4+\pi$

$\Theta =5\pi /4$ or $9\pi /4$

The solution set for the trigonometric equation $\text{tan }\theta =1$

$=\{-3\pi /4,\pi /4,-7\pi /4,-11\pi /4,5\pi /4,9\pi /4\}$