Trigonometry Examples

$3 \tan (x) - 1 = \tan (x)$

Step 1

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

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Step 1.1

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

$3 \tan (x) - 1 - \tan (x) = 0$

Step 1.2

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

$2 \tan (x) - 1 = 0$

Step 2

Add $1$ to both sides of the equation.

$2 \tan (x) = 1$

Step 3

Divide each term in $2 \tan (x) = 1$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \tan (x) = 1$ by $2$ .

$\frac{2 \tan (x)}{2} = \frac{1}{2}$

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.

$\frac{2 \tan (x)}{2} = \frac{1}{2}$

Step 3.2.1.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{1}{2}$

Step 4

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

$x = \arctan (\frac{1}{2})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arctan (\frac{1}{2})$ .

$x = 0.4636476$

Step 6

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

$x = (3.14159265) + 0.4636476$

Step 7

Solve for $x$ .

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Step 7.1

Remove parentheses.

$x = 3.14159265 + 0.4636476$

Step 7.2

Remove parentheses.

$x = (3.14159265) + 0.4636476$

Step 7.3

Add $3.14159265$ and $0.4636476$ .

$x = 3.60524026$

Step 8

Find the period of $\tan (x)$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| 1 |}$

Step 8.3

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

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 10

Consolidate $0.4636476 + π n$ and $3.60524026 + π n$ to $0.4636476 + π n$ .

$x = 0.4636476 + π n$ , for any integer $n$