Trigonometry Examples

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

Step 1

Divide each term in $2 \cos^{2} (2 x) = 1$ by $2$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $2 \cos^{2} (2 x) = 1$ by $2$ .

$\frac{2 \cos^{2} (2 x)}{2} = \frac{1}{2}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cos^{2} (2 x)}{2} = \frac{1}{2}$

Step 1.2.1.2

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

$\cos^{2} (2 x) = \frac{1}{2}$

Step 2

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

$\cos (2 x) = \pm \sqrt{\frac{1}{2}}$

Step 3

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Tap for more steps...

Step 3.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\cos (2 x) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 3.2

Any root of $1$ is $1$ .

$\cos (2 x) = \pm \frac{1}{\sqrt{2}}$

Step 3.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (2 x) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (2 x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (2 x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (2 x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos (2 x) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (2 x) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.6.4.1

Cancel the common factor.

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

Step 3.4.6.4.2

Rewrite the expression.

$\cos (2 x) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 3.4.6.5

Evaluate the exponent.

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$2 x = \frac{π}{4}$

Step 6.3

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

Tap for more steps...

Step 6.3.1

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

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

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor.

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

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{2}$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{2}$

Step 6.3.3.2

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

Tap for more steps...

Step 6.3.3.2.1

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

$x = \frac{π}{4 \cdot 2}$

Step 6.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

Step 6.4

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.

$2 x = 2 π - \frac{π}{4}$

Step 6.5

Solve for $x$ .

Tap for more steps...

Step 6.5.1

Simplify.

Tap for more steps...

Step 6.5.1.1

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

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

Step 6.5.1.2

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 6.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 4 - π}{4}$

Step 6.5.1.4

Multiply $4$ by $2$ .

$2 x = \frac{8 π - π}{4}$

Step 6.5.1.5

Subtract $π$ from $8 π$ .

$2 x = \frac{7 π}{4}$

Step 6.5.2

Divide each term in $2 x = \frac{7 π}{4}$ by $2$ and simplify.

Tap for more steps...

Step 6.5.2.1

Divide each term in $2 x = \frac{7 π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2}$

Step 6.5.2.2

Simplify the left side.

Tap for more steps...

Step 6.5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2}$

Step 6.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{4}}{2}$

Step 6.5.2.3

Simplify the right side.

Tap for more steps...

Step 6.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{4} \cdot \frac{1}{2}$

Step 6.5.2.3.2

Multiply $\frac{7 π}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 6.5.2.3.2.1

Multiply $\frac{7 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{7 π}{4 \cdot 2}$

Step 6.5.2.3.2.2

Multiply $4$ by $2$ .

$x = \frac{7 π}{8}$

Step 6.6

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

Tap for more steps...

Step 6.6.1

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

$\frac{2 π}{| b |}$

Step 6.6.2

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

$\frac{2 π}{| 2 |}$

Step 6.6.3

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

$\frac{2 π}{2}$

Step 6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.6.4.2

Divide $π$ by $1$ .

$π$

Step 6.7

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

$x = \frac{π}{8} + π n, \frac{7 π}{8} + π n$ , for any integer $n$

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$2 x = \frac{3 π}{4}$

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.2.1.1

Cancel the common factor.

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

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{4}}{2}$

Step 7.3.3

Simplify the right side.

Tap for more steps...

Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{4} \cdot \frac{1}{2}$

Step 7.3.3.2

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

Tap for more steps...

Step 7.3.3.2.1

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

$x = \frac{3 π}{4 \cdot 2}$

Step 7.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{3 π}{8}$

Step 7.4

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.

$2 x = 2 π - \frac{3 π}{4}$

Step 7.5

Solve for $x$ .

Tap for more steps...

Step 7.5.1

Simplify.

Tap for more steps...

Step 7.5.1.1

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

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

Step 7.5.1.2

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 7.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 7.5.1.4

Multiply $4$ by $2$ .

$2 x = \frac{8 π - 3 π}{4}$

Step 7.5.1.5

Subtract $3 π$ from $8 π$ .

$2 x = \frac{5 π}{4}$

Step 7.5.2

Divide each term in $2 x = \frac{5 π}{4}$ by $2$ and simplify.

Tap for more steps...

Step 7.5.2.1

Divide each term in $2 x = \frac{5 π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 π}{4}}{2}$

Step 7.5.2.2

Simplify the left side.

Tap for more steps...

Step 7.5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 π}{4}}{2}$

Step 7.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{4}}{2}$

Step 7.5.2.3

Simplify the right side.

Tap for more steps...

Step 7.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{4} \cdot \frac{1}{2}$

Step 7.5.2.3.2

Multiply $\frac{5 π}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 7.5.2.3.2.1

Multiply $\frac{5 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{4 \cdot 2}$

Step 7.5.2.3.2.2

Multiply $4$ by $2$ .

$x = \frac{5 π}{8}$

Step 7.6

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

Tap for more steps...

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 7.6.3

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

$\frac{2 π}{2}$

Step 7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.6.4.2

Divide $π$ by $1$ .

$π$

Step 7.7

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

$x = \frac{3 π}{8} + π n, \frac{5 π}{8} + π n$ , for any integer $n$

Step 8

List all of the solutions.

$x = \frac{π}{8} + π n, \frac{7 π}{8} + π n, \frac{3 π}{8} + π n, \frac{5 π}{8} + π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{4}$ , for any integer $n$