Precalculus Examples

sin (x) = cos (x)

$\sin (x) = \cos (x)$

Step 1

Divide each term in the equation by

cos (x)

$\cos (x)$ .

\frac{sin (x)}{cos (x)} = \frac{cos (x)}{cos (x)}

$\frac{\sin (x)}{\cos (x)} = \frac{\cos (x)}{\cos (x)}$

Step 2

Convert from

\frac{sin (x)}{cos (x)}

$\frac{\sin (x)}{\cos (x)}$ to

tan (x)

$\tan (x)$ .

tan (x) = \frac{cos (x)}{cos (x)}

$\tan (x) = \frac{\cos (x)}{\cos (x)}$

Step 3

Cancel the common factor of

cos (x)

$\cos (x)$ .

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Step 3.1

Cancel the common factor.

$\tan (x) = \frac{\cos (x)}{\cos (x)}$

Step 3.2

Rewrite the expression.

$\tan (x) = 1$

Step 4

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

$x$ from inside the tangent.

$x = \arctan (1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of

$\arctan (1)$ is

$\frac{π}{4}$ .

$x = \frac{π}{4}$

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.

$x = π + \frac{π}{4}$

Step 7

Simplify

$π + \frac{π}{4}$ .

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Step 7.1

To write

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

$\frac{4}{4}$ .

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

Step 7.2

Combine fractions.

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Step 7.2.1

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 7.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Move

$4$ to the left of

$π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 7.3.2

Add

$4 π$ and

$π$ .

$x = \frac{5 π}{4}$

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.

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

$n$

Step 10

Consolidate the answers.

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

$n$