Trigonometry Examples

$\sin^{2} (5 x) = - \frac{3}{4}$

Step 1

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

$\sin (5 x) = \pm \sqrt{- \frac{3}{4}}$

Step 2

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

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Step 2.1

Rewrite $- \frac{3}{4}$ as ${(\frac{i}{2})}^{2} \cdot 3$ .

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Step 2.1.1

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

$\sin (5 x) = \pm \sqrt{i^{2} \frac{3}{4}}$

Step 2.1.2

Factor the perfect power $1^{2}$ out of $3$ .

$\sin (5 x) = \pm \sqrt{i^{2} \frac{1^{2} \cdot 3}{4}}$

Step 2.1.3

Factor the perfect power $2^{2}$ out of $4$ .

$\sin (5 x) = \pm \sqrt{i^{2} \frac{1^{2} \cdot 3}{2^{2} \cdot 1}}$

Step 2.1.4

Rearrange the fraction $\frac{1^{2} \cdot 3}{2^{2} \cdot 1}$ .

$\sin (5 x) = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 3)}$

Step 2.1.5

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

$\sin (5 x) = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 3}$

Step 2.2

Pull terms out from under the radical.

$\sin (5 x) = \pm \frac{i}{2} \sqrt{3}$

Step 2.3

Combine $\frac{i}{2}$ and $\sqrt{3}$ .

$\sin (5 x) = \pm \frac{i \sqrt{3}}{2}$

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.

$\sin (5 x) = \frac{i \sqrt{3}}{2}$

Step 3.2

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

$\sin (5 x) = - \frac{i \sqrt{3}}{2}$

Step 3.3

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

$\sin (5 x) = \frac{i \sqrt{3}}{2}, - \frac{i \sqrt{3}}{2}$

Step 4

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

$\sin (5 x) = \frac{i \sqrt{3}}{2}$

$\sin (5 x) = - \frac{i \sqrt{3}}{2}$

Step 5

Solve for $x$ in $\sin (5 x) = \frac{i \sqrt{3}}{2}$ .

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Step 5.1

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

$5 x = \arcsin (\frac{i \sqrt{3}}{2})$

Step 5.2

The inverse sine of $\arcsin (\frac{i \sqrt{3}}{2})$ is undefined.

Undefined

Step 6

Solve for $x$ in $\sin (5 x) = - \frac{i \sqrt{3}}{2}$ .

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Step 6.1

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

$5 x = \arcsin (- \frac{i \sqrt{3}}{2})$

Step 6.2

The inverse sine of $\arcsin (- \frac{i \sqrt{3}}{2})$ is undefined.

Undefined

Step 7

List all of the solutions.

No solution