Trigonometry Examples

$3 \tan (θ) - 2 = \tan (θ)$

Step 1

Move all terms containing $\tan (θ)$ to the left side of the equation.

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Step 1.1

Subtract $\tan (θ)$ from both sides of the equation.

$3 \tan (θ) - 2 - \tan (θ) = 0$

Step 1.2

Subtract $\tan (θ)$ from $3 \tan (θ)$ .

$2 \tan (θ) - 2 = 0$

Step 2

Add $2$ to both sides of the equation.

$2 \tan (θ) = 2$

Step 3

Divide each term in $2 \tan (θ) = 2$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \tan (θ) = 2$ by $2$ .

$\frac{2 \tan (θ)}{2} = \frac{2}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 \tan (θ)}{2} = \frac{2}{2}$

Step 3.2.1.2

Divide $\tan (θ)$ by $1$ .

$\tan (θ) = \frac{2}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $2$ by $2$ .

$\tan (θ) = 1$

Step 4

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

$θ = \arctan (1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arctan (1)$ is $45$ .

$θ = 45$

Step 6

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

$θ = 180 + 45$

Step 7

Add $180$ and $45$ .

$θ = 225$

Step 8

Find the period of $\tan (θ)$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

$\frac{180}{1}$

Step 8.4

Divide $180$ by $1$ .

$180$

Step 9

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

$θ = 45 + 180 n, 225 + 180 n$ , for any integer $n$

Step 10

Consolidate the answers.

$θ = 45 + 180 n$ , for any integer $n$