Precalculus Examples

$\sin (x) + \cos (x) = 0$

Step 1

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{\cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3

Cancel the common factor of $\cos (x)$ .

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Step 3.1

Cancel the common factor.

$\tan (x) + \frac{\cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.2

Rewrite the expression.

$\tan (x) + 1 = \frac{0}{\cos (x)}$

Step 4

Separate fractions.

$\tan (x) + 1 = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + 1 = \frac{0}{1} \cdot \sec (x)$

Step 6

Divide $0$ by $1$ .

$\tan (x) + 1 = 0 \sec (x)$

Step 7

Multiply $0$ by $\sec (x)$ .

$\tan (x) + 1 = 0$

Step 8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 9

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

$x = \arctan (- 1)$

Step 10

Simplify the right side.

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Step 10.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 11

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

$x = - \frac{π}{4} - π$

Step 12

Simplify the expression to find the second solution.

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Step 12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 12.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 13

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

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Step 13.1

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

$\frac{π}{| b |}$

Step 13.2

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

$\frac{π}{| 1 |}$

Step 13.3

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

$\frac{π}{1}$

Step 13.4

Divide $π$ by $1$ .

$π$

Step 14

Add $π$ to every negative angle to get positive angles.

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Step 14.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 14.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 14.3

Combine fractions.

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Step 14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 14.4

Simplify the numerator.

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Step 14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 15

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

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 16

Consolidate the answers.

$x = \frac{3 π}{4} + π n$ , for any integer $n$