Trigonometry Examples

tan (3 x) = 1

$\tan (3 x) = 1$

Step 1

Take the inverse tangent of both sides of the equation to extract

x

$x$ from inside the tangent.

3 x = arctan (1)

$3 x = \arctan (1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arctan (1)

$\arctan (1)$ is

\frac{π}{4}

$\frac{π}{4}$ .

3 x = \frac{π}{4}

$3 x = \frac{π}{4}$

3 x = \frac{π}{4}

$3 x = \frac{π}{4}$

Step 3

Divide each term in

3 x = \frac{π}{4}

$3 x = \frac{π}{4}$ by

3

$3$ and simplify.

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Step 3.1

Divide each term in

3 x = \frac{π}{4}

$3 x = \frac{π}{4}$ by

3

$3$ .

\frac{3 x}{3} = \frac{\frac{π}{4}}{3}

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

3

$3$ .

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Step 3.2.1.1

Cancel the common factor.

\frac{3 x}{3} = \frac{\frac{π}{4}}{3}

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

Step 3.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\frac{π}{4}}{3}

$x = \frac{\frac{π}{4}}{3}$

x = \frac{\frac{π}{4}}{3}

$x = \frac{\frac{π}{4}}{3}$

x = \frac{\frac{π}{4}}{3}

$x = \frac{\frac{π}{4}}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

x = \frac{π}{4} \cdot \frac{1}{3}

$x = \frac{π}{4} \cdot \frac{1}{3}$

Step 3.3.2

Multiply

\frac{π}{4} \cdot \frac{1}{3}

$\frac{π}{4} \cdot \frac{1}{3}$ .

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Step 3.3.2.1

Multiply

\frac{π}{4}

$\frac{π}{4}$ by

\frac{1}{3}

$\frac{1}{3}$ .

x = \frac{π}{4 \cdot 3}

$x = \frac{π}{4 \cdot 3}$

Step 3.3.2.2

Multiply

4

$4$ by

3

$3$ .

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

Step 4

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

π

$π$ to find the solution in the fourth quadrant.

3 x = π + \frac{π}{4}

$3 x = π + \frac{π}{4}$

Step 5

Solve for

x

$x$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write

π

$π$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

3 x = π \cdot \frac{4}{4} + \frac{π}{4}

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

Step 5.1.2

Combine

π

$π$ and

\frac{4}{4}

$\frac{4}{4}$ .

3 x = \frac{π \cdot 4}{4} + \frac{π}{4}

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

Step 5.1.3

Combine the numerators over the common denominator.

3 x = \frac{π \cdot 4 + π}{4}

$3 x = \frac{π \cdot 4 + π}{4}$

Step 5.1.4

Add

π \cdot 4

$π \cdot 4$ and

π

$π$ .

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Step 5.1.4.1

Reorder

π

$π$ and

4

$4$ .

3 x = \frac{4 \cdot π + π}{4}

$3 x = \frac{4 \cdot π + π}{4}$

Step 5.1.4.2

Add

4 \cdot π

$4 \cdot π$ and

π

$π$ .

3 x = \frac{5 \cdot π}{4}

$3 x = \frac{5 \cdot π}{4}$

3 x = \frac{5 \cdot π}{4}

$3 x = \frac{5 \cdot π}{4}$

3 x = \frac{5 \cdot π}{4}

$3 x = \frac{5 \cdot π}{4}$

Step 5.2

Divide each term in

3 x = \frac{5 \cdot π}{4}

$3 x = \frac{5 \cdot π}{4}$ by

3

$3$ and simplify.

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Step 5.2.1

Divide each term in

3 x = \frac{5 \cdot π}{4}

$3 x = \frac{5 \cdot π}{4}$ by

3

$3$ .

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

3

$3$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

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

Step 5.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\frac{5 \cdot π}{4}}{3}

$x = \frac{\frac{5 \cdot π}{4}}{3}$

x = \frac{\frac{5 \cdot π}{4}}{3}

$x = \frac{\frac{5 \cdot π}{4}}{3}$

x = \frac{\frac{5 \cdot π}{4}}{3}

$x = \frac{\frac{5 \cdot π}{4}}{3}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 5.2.3.2

Multiply

\frac{5 π}{4} \cdot \frac{1}{3}

$\frac{5 π}{4} \cdot \frac{1}{3}$ .

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Step 5.2.3.2.1

Multiply

\frac{5 π}{4}

$\frac{5 π}{4}$ by

\frac{1}{3}

$\frac{1}{3}$ .

x = \frac{5 π}{4 \cdot 3}

$x = \frac{5 π}{4 \cdot 3}$

Step 5.2.3.2.2

Multiply

4

$4$ by

3

$3$ .

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

Step 6

Find the period of

tan (3 x)

$\tan (3 x)$ .

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Step 6.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 6.2

Replace

b

$b$ with

3

$3$ in the formula for period.

\frac{π}{| 3 |}

$\frac{π}{| 3 |}$

Step 6.3

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

0

$0$ and

3

$3$ is

3

$3$ .

\frac{π}{3}

$\frac{π}{3}$

\frac{π}{3}

$\frac{π}{3}$

Step 7

The period of the

tan (3 x)

$\tan (3 x)$ function is

\frac{π}{3}

$\frac{π}{3}$ so values will repeat every

\frac{π}{3}

$\frac{π}{3}$ radians in both directions.

x = \frac{π}{12} + \frac{π n}{3}, \frac{5 π}{12} + \frac{π n}{3}

$x = \frac{π}{12} + \frac{π n}{3}, \frac{5 π}{12} + \frac{π n}{3}$ , for any integer

$n$

Step 8

Consolidate the answers.

$x = \frac{π}{12} + \frac{π n}{3}$ , for any integer

$n$