Trigonometry Examples

$\sin (2 x) + \cos (2 x) = 0$

Step 1

Divide each term in the equation by $\cos (2 x)$ .

$\frac{\sin (2 x)}{\cos (2 x)} + \frac{\cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 2

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\tan (2 x) + \frac{\cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 3

Cancel the common factor of $\cos (2 x)$ .

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Step 3.1

Cancel the common factor.

$\tan (2 x) + \frac{\cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 3.2

Rewrite the expression.

$\tan (2 x) + 1 = \frac{0}{\cos (2 x)}$

Step 4

Separate fractions.

$\tan (2 x) + 1 = \frac{0}{1} \cdot \frac{1}{\cos (2 x)}$

Step 5

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\tan (2 x) + 1 = \frac{0}{1} \cdot \sec (2 x)$

Step 6

Divide $0$ by $1$ .

$\tan (2 x) + 1 = 0 \sec (2 x)$

Step 7

Multiply $0$ by $\sec (2 x)$ .

$\tan (2 x) + 1 = 0$

Step 8

Subtract $1$ from both sides of the equation.

$\tan (2 x) = - 1$

Step 9

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

$2 x = \arctan (- 1)$

Step 10

Simplify the right side.

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Step 10.1

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

$2 x = - \frac{π}{4}$

Step 11

Divide each term in $2 x = - \frac{π}{4}$ by $2$ and simplify.

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Step 11.1

Divide each term in $2 x = - \frac{π}{4}$ by $2$ .

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

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $2$ .

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Step 11.2.1.1

Cancel the common factor.

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

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{4}}{2}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{4} \cdot \frac{1}{2}$

Step 11.3.2

Multiply $- \frac{π}{4} \cdot \frac{1}{2}$ .

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Step 11.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

$x = - \frac{π}{2 \cdot 4}$

Step 11.3.2.2

Multiply $2$ by $4$ .

$x = - \frac{π}{8}$

Step 12

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.

$2 x = - \frac{π}{4} - π$

Step 13

Simplify the expression to find the second solution.

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Step 13.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$2 x = - \frac{π}{4} - π + 2 π$

Step 13.2

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

$2 x = \frac{3 π}{4}$

Step 13.3

Divide each term in $2 x = \frac{3 π}{4}$ by $2$ and simplify.

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Step 13.3.1

Divide each term in $2 x = \frac{3 π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{4}}{2}$

Step 13.3.2

Simplify the left side.

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Step 13.3.2.1

Cancel the common factor of $2$ .

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Step 13.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{4}}{2}$

Step 13.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{4}}{2}$

Step 13.3.3

Simplify the right side.

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Step 13.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{4} \cdot \frac{1}{2}$

Step 13.3.3.2

Multiply $\frac{3 π}{4} \cdot \frac{1}{2}$ .

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Step 13.3.3.2.1

Multiply $\frac{3 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{4 \cdot 2}$

Step 13.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{3 π}{8}$

Step 14

Find the period of $\tan (2 x)$ .

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Step 14.1

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

$\frac{π}{| b |}$

Step 14.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 14.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 15

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Step 15.1

Add $\frac{π}{2}$ to $- \frac{π}{8}$ to find the positive angle.

$- \frac{π}{8} + \frac{π}{2}$

Step 15.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π}{2} \cdot \frac{4}{4} - \frac{π}{8}$

Step 15.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.3.1

Multiply $\frac{π}{2}$ by $\frac{4}{4}$ .

$\frac{π \cdot 4}{2 \cdot 4} - \frac{π}{8}$

Step 15.3.2

Multiply $2$ by $4$ .

$\frac{π \cdot 4}{8} - \frac{π}{8}$

Step 15.4

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{8}$

Step 15.5

Simplify the numerator.

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Step 15.5.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{8}$

Step 15.5.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{8}$

Step 15.6

List the new angles.

$x = \frac{3 π}{8}$

Step 16

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{3 π}{8} + \frac{π n}{2}, \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 17

Consolidate the answers.

$x = \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$