Calculus Examples

$\tan (2 x) = 1$

Step 1

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

$x$ from inside the tangent.

$2 x = \arctan (1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

$\arctan (1)$ is

$\frac{π}{4}$ .

$2 x = \frac{π}{4}$

Step 3

Divide each term in

$2 x = \frac{π}{4}$ by

$2$ and simplify.

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Step 3.1

Divide each term in

$2 x = \frac{π}{4}$ by

$2$ .

$\frac{2 x}{2} = \frac{\frac{π}{4}}{2}$

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.

$\frac{2 x}{2} = \frac{\frac{π}{4}}{2}$

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{4}}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{2}$

Step 3.3.2

Multiply

$\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 3.3.2.1

Multiply

$\frac{π}{4}$ by

$\frac{1}{2}$ .

$x = \frac{π}{4 \cdot 2}$

Step 3.3.2.2

Multiply

$4$ by

$2$ .

$x = \frac{π}{8}$

Step 4

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.

$2 x = π + \frac{π}{4}$

Step 5

Solve for

$x$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write

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

$\frac{4}{4}$ .

$2 x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 5.1.2

Combine

$π$ and

$\frac{4}{4}$ .

$2 x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 4 + π}{4}$

Step 5.1.4

Add

$π \cdot 4$ and

$π$ .

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Step 5.1.4.1

Reorder

$π$ and

$4$ .

$2 x = \frac{4 \cdot π + π}{4}$

Step 5.1.4.2

Add

$4 \cdot π$ and

$π$ .

$2 x = \frac{5 \cdot π}{4}$

Step 5.2

Divide each term in

$2 x = \frac{5 \cdot π}{4}$ by

$2$ and simplify.

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Step 5.2.1

Divide each term in

$2 x = \frac{5 \cdot π}{4}$ by

$2$ .

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{4}}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{4}}{2}$

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{5 \cdot π}{4}}{2}$

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.

$x = \frac{5 \cdot π}{4} \cdot \frac{1}{2}$

Step 5.2.3.2

Multiply

$\frac{5 π}{4} \cdot \frac{1}{2}$ .

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Step 5.2.3.2.1

Multiply

$\frac{5 π}{4}$ by

$\frac{1}{2}$ .

$x = \frac{5 π}{4 \cdot 2}$

Step 5.2.3.2.2

Multiply

$4$ by

$2$ .

$x = \frac{5 π}{8}$

Step 6

Find the period of

$\tan (2 x)$ .

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Step 6.1

The period of the function can be calculated using

$\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 6.2

Replace

$b$ with

$2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 6.3

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

$0$ and

$2$ is

$2$ .

$\frac{π}{2}$

Step 7

The period of the

$\tan (2 x)$ function is

$\frac{π}{2}$ so values will repeat every

$\frac{π}{2}$ radians in both directions.

$x = \frac{π}{8} + \frac{π n}{2}, \frac{5 π}{8} + \frac{π n}{2}$ , for any integer

$n$

Step 8

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{2}$ , for any integer

$n$