Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify $\pm \sqrt{- \frac{\sqrt{3}}{2}}$ .

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Step 2.1

Rewrite $- 1$ as $i^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 2.3

Rewrite $\sqrt{\frac{\sqrt{3}}{2}}$ as $\frac{\sqrt{\sqrt{3}}}{\sqrt{2}}$ .

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

Step 2.4

Rewrite $\sqrt{\sqrt{3}}$ as $\sqrt[4]{3}$ .

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

Step 2.5

Multiply $\frac{\sqrt[4]{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \pm i (\frac{\sqrt[4]{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 2.6

Combine and simplify the denominator.

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Step 2.6.1

Multiply $\frac{\sqrt[4]{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.6.2

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

$\cos (x) = \pm i \frac{\sqrt[4]{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.6.3

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

$\cos (x) = \pm i \frac{\sqrt[4]{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.6.4

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

$\cos (x) = \pm i \frac{\sqrt[4]{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.6.5

Add $1$ and $1$ .

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

Step 2.6.6

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

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Step 2.6.6.1

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

$\cos (x) = \pm i \frac{\sqrt[4]{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.6.6.2

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

$\cos (x) = \pm i \frac{\sqrt[4]{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.6.6.3

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

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

Step 2.6.6.4

Cancel the common factor of $2$ .

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Step 2.6.6.4.1

Cancel the common factor.

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

Step 2.6.6.4.2

Rewrite the expression.

$\cos (x) = \pm i \frac{\sqrt[4]{3} \sqrt{2}}{2^{1}}$

Step 2.6.6.5

Evaluate the exponent.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Rewrite the expression using the least common index of $4$ .

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Step 2.7.1.1

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

$\cos (x) = \pm i \frac{\sqrt[4]{3} \cdot 2^{\frac{1}{2}}}{2}$

Step 2.7.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

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

Step 2.7.1.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

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

Step 2.7.2

Combine using the product rule for radicals.

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

Step 2.7.3

Raise $2$ to the power of $2$ .

$\cos (x) = \pm i \frac{\sqrt[4]{3 \cdot 4}}{2}$

Step 2.8

Combine fractions.

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Step 2.8.1

Multiply $3$ by $4$ .

$\cos (x) = \pm i \frac{\sqrt[4]{12}}{2}$

Step 2.8.2

Combine $i$ and $\frac{\sqrt[4]{12}}{2}$ .

$\cos (x) = \pm \frac{i \sqrt[4]{12}}{2}$

Step 3

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

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Step 3.1

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

$\cos (x) = \frac{i \sqrt[4]{12}}{2}$

Step 3.2

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

$\cos (x) = - \frac{i \sqrt[4]{12}}{2}$

Step 3.3

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

$\cos (x) = \frac{i \sqrt[4]{12}}{2}, - \frac{i \sqrt[4]{12}}{2}$

Step 4

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

$\cos (x) = \frac{i \sqrt[4]{12}}{2}$

$\cos (x) = - \frac{i \sqrt[4]{12}}{2}$

Step 5

Solve for $x$ in $\cos (x) = \frac{i \sqrt[4]{12}}{2}$ .

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Step 5.1

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

$x = \arccos (\frac{i \sqrt[4]{12}}{2})$

Step 5.2

The inverse cosine of $\arccos (\frac{i \sqrt[4]{12}}{2})$ is undefined.

Undefined

Step 6

Solve for $x$ in $\cos (x) = - \frac{i \sqrt[4]{12}}{2}$ .

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Step 6.1

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

$x = \arccos (- \frac{i \sqrt[4]{12}}{2})$

Step 6.2

The inverse cosine of $\arccos (- \frac{i \sqrt[4]{12}}{2})$ is undefined.

Undefined

Step 7

List all of the solutions.

No solution