Trigonometry Examples

$2 \cos^{2} (2 x) - 1 = \sin (4 x)$

Step 1

Subtract $\sin (4 x)$ from both sides of the equation.

$2 \cos^{2} (2 x) - 1 - \sin (4 x) = 0$

Step 2

Simplify $2 \cos^{2} (2 x) - 1 - \sin (4 x)$ .

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Step 2.1

Apply the cosine double-angle identity.

$\cos (2 (2 x)) - \sin (4 x) = 0$

Step 2.2

Multiply $2$ by $2$ .

$\cos (4 x) - \sin (4 x) = 0$

Step 3

Divide each term in the equation by $\cos (4 x)$ .

$\frac{\cos (4 x)}{\cos (4 x)} + \frac{- \sin (4 x)}{\cos (4 x)} = \frac{0}{\cos (4 x)}$

Step 4

Cancel the common factor of $\cos (4 x)$ .

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Step 4.1

Cancel the common factor.

$\frac{\cos (4 x)}{\cos (4 x)} + \frac{- \sin (4 x)}{\cos (4 x)} = \frac{0}{\cos (4 x)}$

Step 4.2

Rewrite the expression.

$1 + \frac{- \sin (4 x)}{\cos (4 x)} = \frac{0}{\cos (4 x)}$

Step 5

Separate fractions.

$1 + \frac{- 1}{1} \cdot \frac{\sin (4 x)}{\cos (4 x)} = \frac{0}{\cos (4 x)}$

Step 6

Convert from $\frac{\sin (4 x)}{\cos (4 x)}$ to $\tan (4 x)$ .

$1 + \frac{- 1}{1} \cdot \tan (4 x) = \frac{0}{\cos (4 x)}$

Step 7

Divide $- 1$ by $1$ .

$1 - \tan (4 x) = \frac{0}{\cos (4 x)}$

Step 8

Separate fractions.

$1 - \tan (4 x) = \frac{0}{1} \cdot \frac{1}{\cos (4 x)}$

Step 9

Convert from $\frac{1}{\cos (4 x)}$ to $\sec (4 x)$ .

$1 - \tan (4 x) = \frac{0}{1} \cdot \sec (4 x)$

Step 10

Divide $0$ by $1$ .

$1 - \tan (4 x) = 0 \sec (4 x)$

Step 11

Multiply $0$ by $\sec (4 x)$ .

$1 - \tan (4 x) = 0$

Step 12

Subtract $1$ from both sides of the equation.

$- \tan (4 x) = - 1$

Step 13

Divide each term in $- \tan (4 x) = - 1$ by $- 1$ and simplify.

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Step 13.1

Divide each term in $- \tan (4 x) = - 1$ by $- 1$ .

$\frac{- \tan (4 x)}{- 1} = \frac{- 1}{- 1}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (4 x)}{1} = \frac{- 1}{- 1}$

Step 13.2.2

Divide $\tan (4 x)$ by $1$ .

$\tan (4 x) = \frac{- 1}{- 1}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Divide $- 1$ by $- 1$ .

$\tan (4 x) = 1$

Step 14

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

$4 x = \arctan (1)$

Step 15

Simplify the right side.

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Step 15.1

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

$4 x = \frac{π}{4}$

Step 16

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

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Step 16.1

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

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

Step 16.2

Simplify the left side.

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Step 16.2.1

Cancel the common factor of $4$ .

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Step 16.2.1.1

Cancel the common factor.

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

Step 16.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{4}$

Step 16.3

Simplify the right side.

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Step 16.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{4}$

Step 16.3.2

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

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Step 16.3.2.1

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

$x = \frac{π}{4 \cdot 4}$

Step 16.3.2.2

Multiply $4$ by $4$ .

$x = \frac{π}{16}$

Step 17

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.

$4 x = π + \frac{π}{4}$

Step 18

Solve for $x$ .

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Step 18.1

Simplify.

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Step 18.1.1

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

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

Step 18.1.2

Combine $π$ and $\frac{4}{4}$ .

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

Step 18.1.3

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 4 + π}{4}$

Step 18.1.4

Add $π \cdot 4$ and $π$ .

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Step 18.1.4.1

Reorder $π$ and $4$ .

$4 x = \frac{4 \cdot π + π}{4}$

Step 18.1.4.2

Add $4 \cdot π$ and $π$ .

$4 x = \frac{5 \cdot π}{4}$

Step 18.2

Divide each term in $4 x = \frac{5 \cdot π}{4}$ by $4$ and simplify.

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Step 18.2.1

Divide each term in $4 x = \frac{5 \cdot π}{4}$ by $4$ .

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

Step 18.2.2

Simplify the left side.

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Step 18.2.2.1

Cancel the common factor of $4$ .

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Step 18.2.2.1.1

Cancel the common factor.

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

Step 18.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{4}}{4}$

Step 18.2.3

Simplify the right side.

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Step 18.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 18.2.3.2

Multiply $\frac{5 π}{4} \cdot \frac{1}{4}$ .

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Step 18.2.3.2.1

Multiply $\frac{5 π}{4}$ by $\frac{1}{4}$ .

$x = \frac{5 π}{4 \cdot 4}$

Step 18.2.3.2.2

Multiply $4$ by $4$ .

$x = \frac{5 π}{16}$

Step 19

Find the period of $\tan (4 x)$ .

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Step 19.1

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

$\frac{π}{| b |}$

Step 19.2

Replace $b$ with $4$ in the formula for period.

$\frac{π}{| 4 |}$

Step 19.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{π}{4}$

Step 20

The period of the $\tan (4 x)$ function is $\frac{π}{4}$ so values will repeat every $\frac{π}{4}$ radians in both directions.

$x = \frac{π}{16} + \frac{π n}{4}, \frac{5 π}{16} + \frac{π n}{4}$ , for any integer $n$

Step 21

Consolidate the answers.

$x = \frac{π}{16} + \frac{π n}{4}$ , for any integer $n$