Trigonometry Examples

$3 \tan (θ) + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$3 \tan (θ) = - 1$

Step 2

Divide each term in $3 \tan (θ) = - 1$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \tan (θ) = - 1$ by $3$ .

$\frac{3 \tan (θ)}{3} = \frac{- 1}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 \tan (θ)}{3} = \frac{- 1}{3}$

Step 2.2.1.2

Divide $\tan (θ)$ by $1$ .

$\tan (θ) = \frac{- 1}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\tan (θ) = - \frac{1}{3}$

Step 3

Take the inverse tangent of both sides of the equation to extract $θ$ from inside the tangent.

$θ = \arctan (- \frac{1}{3})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (- \frac{1}{3})$ .

$θ = - 18.43494882$

Step 5

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the third quadrant.

$θ = - 18.43494882 - 180$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $360 °$ to $- 18.43494882 - 180 °$ .

$θ = - 18.43494882 - 180 ° + 360 °$

Step 6.2

The resulting angle of $161.56505117 °$ is positive and coterminal with $- 18.43494882 - 180$ .

$θ = 161.56505117 °$

Step 7

Find the period of $\tan (θ)$ .

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Step 7.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{180}{1}$

Step 7.4

Divide $180$ by $1$ .

$180$

Step 8

Add $180$ to every negative angle to get positive angles.

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Step 8.1

Add $180$ to $- 18.43494882$ to find the positive angle.

$- 18.43494882 + 180$

Step 8.2

Subtract $18.43494882$ from $180$ .

$161.56505117$

Step 8.3

List the new angles.

$θ = 161.56505117$

Step 9

The period of the $\tan (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 161.56505117 + 180 n, 161.56505117 + 180 n$ , for any integer $n$

Step 10

Consolidate the answers.

$θ = 161.56505117 + 180 n$ , for any integer $n$