Trigonometry Examples

$\tan (5 x) - \sqrt{3} = 0$

Step 1

Add $\sqrt{3}$ to both sides of the equation.

$\tan (5 x) = \sqrt{3}$

Step 2

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

$5 x = \arctan (\sqrt{3})$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arctan (\sqrt{3})$ is $\frac{π}{3}$ .

$5 x = \frac{π}{3}$

Step 4

Divide each term in $5 x = \frac{π}{3}$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 x = \frac{π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{π}{3}}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{π}{3}}{5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{5}$

Step 4.3.2

Multiply $\frac{π}{3} \cdot \frac{1}{5}$ .

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Step 4.3.2.1

Multiply $\frac{π}{3}$ by $\frac{1}{5}$ .

$x = \frac{π}{3 \cdot 5}$

Step 4.3.2.2

Multiply $3$ by $5$ .

$x = \frac{π}{15}$

Step 5

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.

$5 x = π + \frac{π}{3}$

Step 6

Solve for $x$ .

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Step 6.1

Simplify.

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Step 6.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 x = π \cdot \frac{3}{3} + \frac{π}{3}$

Step 6.1.2

Combine $π$ and $\frac{3}{3}$ .

$5 x = \frac{π \cdot 3}{3} + \frac{π}{3}$

Step 6.1.3

Combine the numerators over the common denominator.

$5 x = \frac{π \cdot 3 + π}{3}$

Step 6.1.4

Add $π \cdot 3$ and $π$ .

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Step 6.1.4.1

Reorder $π$ and $3$ .

$5 x = \frac{3 \cdot π + π}{3}$

Step 6.1.4.2

Add $3 \cdot π$ and $π$ .

$5 x = \frac{4 \cdot π}{3}$

Step 6.2

Divide each term in $5 x = \frac{4 \cdot π}{3}$ by $5$ and simplify.

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Step 6.2.1

Divide each term in $5 x = \frac{4 \cdot π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{4 \cdot π}{3}}{5}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $5$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{4 \cdot π}{3}}{5}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{4 \cdot π}{3}}{5}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{4 \cdot π}{3} \cdot \frac{1}{5}$

Step 6.2.3.2

Multiply $\frac{4 π}{3} \cdot \frac{1}{5}$ .

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Step 6.2.3.2.1

Multiply $\frac{4 π}{3}$ by $\frac{1}{5}$ .

$x = \frac{4 π}{3 \cdot 5}$

Step 6.2.3.2.2

Multiply $3$ by $5$ .

$x = \frac{4 π}{15}$

Step 7

Find the period of $\tan (5 x)$ .

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

Replace $b$ with $5$ in the formula for period.

$\frac{π}{| 5 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{π}{5}$

Step 8

The period of the $\tan (5 x)$ function is $\frac{π}{5}$ so values will repeat every $\frac{π}{5}$ radians in both directions.

$x = \frac{π}{15} + \frac{π n}{5}, \frac{4 π}{15} + \frac{π n}{5}$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{15} + \frac{π n}{5}$ , for any integer $n$