Trigonometry Examples

$\sin^{2} (x) + 3 \cos^{2} (x) = 0$

Step 1

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$(1 - \cos^{2} (x)) + 3 \cos^{2} (x) = 0$

Step 2

Add $- \cos^{2} (x)$ and $3 \cos^{2} (x)$ .

$1 + 2 \cos^{2} (x) = 0$

Step 3

Reorder the polynomial.

$2 \cos^{2} (x) + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

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

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Pull terms out from under the radical.

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

Step 7.3

One to any power is one.

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

Step 7.4

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

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

Step 7.5

Any root of $1$ is $1$ .

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

Step 7.6

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

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

Step 7.7

Combine and simplify the denominator.

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Step 7.7.1

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

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

Step 7.7.2

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

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

Step 7.7.3

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

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

Step 7.7.4

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

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

Step 7.7.5

Add $1$ and $1$ .

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

Step 7.7.6

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

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Step 7.7.6.1

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

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

Step 7.7.6.2

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

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

Step 7.7.6.3

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

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

Step 7.7.6.4

Cancel the common factor of $2$ .

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Step 7.7.6.4.1

Cancel the common factor.

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

Step 7.7.6.4.2

Rewrite the expression.

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

Step 7.7.6.5

Evaluate the exponent.

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

Step 7.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

Undefined

Step 11

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

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Step 11.1

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

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

Step 11.2

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

Undefined

Step 12

List all of the solutions.

No solution