Calculus Examples

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Simplify the right side.

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

$2 x = \frac{π}{4}$

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

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{2}$

Simplify the right side.

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Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{2}$

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

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

$x = \frac{π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

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.

$2 x = 2 π - \frac{π}{4}$

Solve for $x$ .

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Simplify.

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

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

Combine $2 π$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 4 - π}{4}$

Multiply $4$ by $2$ .

$2 x = \frac{8 π - π}{4}$

Subtract $π$ from $8 π$ .

$2 x = \frac{7 π}{4}$

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

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{4}}{2}$

Simplify the right side.

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Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{4} \cdot \frac{1}{2}$

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

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

$x = \frac{7 π}{4 \cdot 2}$

Multiply $4$ by $2$ .

$x = \frac{7 π}{8}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 2 |}$

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

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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 $n$