Trigonometry Examples

$4 \cos^{2} (4 x) - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$4 \cos^{2} (4 x) = 3$

Step 2

Divide each term in $4 \cos^{2} (4 x) = 3$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 \cos^{2} (4 x) = 3$ by $4$ .

$\frac{4 \cos^{2} (4 x)}{4} = \frac{3}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 \cos^{2} (4 x)}{4} = \frac{3}{4}$

Step 2.2.1.2

Divide $\cos^{2} (4 x)$ by $1$ .

$\cos^{2} (4 x) = \frac{3}{4}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (4 x) = \pm \sqrt{\frac{3}{4}}$

Step 4

Simplify $\pm \sqrt{\frac{3}{4}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\cos (4 x) = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Rewrite $4$ as $2^{2}$ .

$\cos (4 x) = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (4 x) = \pm \frac{\sqrt{3}}{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$\cos (4 x) = \frac{\sqrt{3}}{2}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (4 x) = - \frac{\sqrt{3}}{2}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (4 x) = \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}$

Step 6

Set up each of the solutions to solve for $x$ .

$\cos (4 x) = \frac{\sqrt{3}}{2}$

$\cos (4 x) = - \frac{\sqrt{3}}{2}$

Step 7

Solve for $x$ in $\cos (4 x) = \frac{\sqrt{3}}{2}$ .

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Step 7.1

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

$4 x = \arccos (\frac{\sqrt{3}}{2})$

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$4 x = \frac{π}{6}$

Step 7.3

Divide each term in $4 x = \frac{π}{6}$ by $4$ and simplify.

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Step 7.3.1

Divide each term in $4 x = \frac{π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $4$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{4}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{4}$

Step 7.3.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{4}$ .

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Step 7.3.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{4}$ .

$x = \frac{π}{6 \cdot 4}$

Step 7.3.3.2.2

Multiply $6$ by $4$ .

$x = \frac{π}{24}$

Step 7.4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$4 x = 2 π - \frac{π}{6}$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Simplify.

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Step 7.5.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$4 x = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.5.1.2

Combine $2 π$ and $\frac{6}{6}$ .

$4 x = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 7.5.1.3

Combine the numerators over the common denominator.

$4 x = \frac{2 π \cdot 6 - π}{6}$

Step 7.5.1.4

Multiply $6$ by $2$ .

$4 x = \frac{12 π - π}{6}$

Step 7.5.1.5

Subtract $π$ from $12 π$ .

$4 x = \frac{11 π}{6}$

Step 7.5.2

Divide each term in $4 x = \frac{11 π}{6}$ by $4$ and simplify.

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Step 7.5.2.1

Divide each term in $4 x = \frac{11 π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{11 π}{6}}{4}$

Step 7.5.2.2

Simplify the left side.

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Step 7.5.2.2.1

Cancel the common factor of $4$ .

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Step 7.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{11 π}{6}}{4}$

Step 7.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{11 π}{6}}{4}$

Step 7.5.2.3

Simplify the right side.

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Step 7.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{11 π}{6} \cdot \frac{1}{4}$

Step 7.5.2.3.2

Multiply $\frac{11 π}{6} \cdot \frac{1}{4}$ .

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Step 7.5.2.3.2.1

Multiply $\frac{11 π}{6}$ by $\frac{1}{4}$ .

$x = \frac{11 π}{6 \cdot 4}$

Step 7.5.2.3.2.2

Multiply $6$ by $4$ .

$x = \frac{11 π}{24}$

Step 7.6

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

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Step 7.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.6.2

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

$\frac{2 π}{| 4 |}$

Step 7.6.3

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

$\frac{2 π}{4}$

Step 7.6.4

Cancel the common factor of $2$ and $4$ .

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Step 7.6.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 7.6.4.2

Cancel the common factors.

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Step 7.6.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 7.6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 7.6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 7.7

The period of the $\cos (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{24} + \frac{π n}{2}, \frac{11 π}{24} + \frac{π n}{2}$ , for any integer $n$

Step 8

Solve for $x$ in $\cos (4 x) = - \frac{\sqrt{3}}{2}$ .

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Step 8.1

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

$4 x = \arccos (- \frac{\sqrt{3}}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$4 x = \frac{5 π}{6}$

Step 8.3

Divide each term in $4 x = \frac{5 π}{6}$ by $4$ and simplify.

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Step 8.3.1

Divide each term in $4 x = \frac{5 π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{5 π}{6}}{4}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $4$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{5 π}{6}}{4}$

Step 8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{6}}{4}$

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{6} \cdot \frac{1}{4}$

Step 8.3.3.2

Multiply $\frac{5 π}{6} \cdot \frac{1}{4}$ .

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Step 8.3.3.2.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{4}$ .

$x = \frac{5 π}{6 \cdot 4}$

Step 8.3.3.2.2

Multiply $6$ by $4$ .

$x = \frac{5 π}{24}$

Step 8.4

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$4 x = 2 π - \frac{5 π}{6}$

Step 8.5

Solve for $x$ .

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Step 8.5.1

Simplify.

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Step 8.5.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$4 x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 8.5.1.2

Combine $2 π$ and $\frac{6}{6}$ .

$4 x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 8.5.1.3

Combine the numerators over the common denominator.

$4 x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 8.5.1.4

Multiply $6$ by $2$ .

$4 x = \frac{12 π - 5 π}{6}$

Step 8.5.1.5

Subtract $5 π$ from $12 π$ .

$4 x = \frac{7 π}{6}$

Step 8.5.2

Divide each term in $4 x = \frac{7 π}{6}$ by $4$ and simplify.

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Step 8.5.2.1

Divide each term in $4 x = \frac{7 π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{7 π}{6}}{4}$

Step 8.5.2.2

Simplify the left side.

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Step 8.5.2.2.1

Cancel the common factor of $4$ .

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Step 8.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{7 π}{6}}{4}$

Step 8.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{4}$

Step 8.5.2.3

Simplify the right side.

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Step 8.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{4}$

Step 8.5.2.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{4}$ .

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Step 8.5.2.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{4}$ .

$x = \frac{7 π}{6 \cdot 4}$

Step 8.5.2.3.2.2

Multiply $6$ by $4$ .

$x = \frac{7 π}{24}$

Step 8.6

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

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Step 8.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.6.2

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

$\frac{2 π}{| 4 |}$

Step 8.6.3

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

$\frac{2 π}{4}$

Step 8.6.4

Cancel the common factor of $2$ and $4$ .

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Step 8.6.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 8.6.4.2

Cancel the common factors.

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Step 8.6.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 8.6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 8.6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 8.7

The period of the $\cos (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{5 π}{24} + \frac{π n}{2}, \frac{7 π}{24} + \frac{π n}{2}$ , for any integer $n$

Step 9

List all of the solutions.

$x = \frac{π}{24} + \frac{π n}{2}, \frac{11 π}{24} + \frac{π n}{2}, \frac{5 π}{24} + \frac{π n}{2}, \frac{7 π}{24} + \frac{π n}{2}$ , for any integer $n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $\frac{π}{24} + \frac{π n}{2}$ and $\frac{7 π}{24} + \frac{π n}{2}$ to $\frac{π}{24} + \frac{π n}{4}$ .

$x = \frac{π}{24} + \frac{π n}{4}, \frac{11 π}{24} + \frac{π n}{2}, \frac{5 π}{24} + \frac{π n}{2}$ , for any integer $n$

Step 10.2

Consolidate $\frac{11 π}{24} + \frac{π n}{2}$ and $\frac{5 π}{24} + \frac{π n}{2}$ to $\frac{5 π}{24} + \frac{π n}{4}$ .

$x = \frac{π}{24} + \frac{π n}{4}, \frac{5 π}{24} + \frac{π n}{4}$ , for any integer $n$