Trigonometry Examples

$4 \sin^{2} (x) = \frac{\sqrt{2}}{2}$

Step 1

Divide each term in $4 \sin^{2} (x) = \frac{\sqrt{2}}{2}$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 \sin^{2} (x) = \frac{\sqrt{2}}{2}$ by $4$ .

$\frac{4 \sin^{2} (x)}{4} = \frac{\frac{\sqrt{2}}{2}}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 \sin^{2} (x)}{4} = \frac{\frac{\sqrt{2}}{2}}{4}$

Step 1.2.1.2

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

$\sin^{2} (x) = \frac{\frac{\sqrt{2}}{2}}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3.2

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

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Step 1.3.2.1

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

$\sin^{2} (x) = \frac{\sqrt{2}}{2 \cdot 4}$

Step 1.3.2.2

Multiply $2$ by $4$ .

$\sin^{2} (x) = \frac{\sqrt{2}}{8}$

Step 2

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

$\sin (x) = \pm \sqrt{\frac{\sqrt{2}}{8}}$

Step 3

Simplify $\pm \sqrt{\frac{\sqrt{2}}{8}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{\sqrt{2}}{8}}$ as $\frac{\sqrt{\sqrt{2}}}{\sqrt{8}}$ .

$\sin (x) = \pm \frac{\sqrt{\sqrt{2}}}{\sqrt{8}}$

Step 3.2

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

$\sin (x) = \pm \frac{\sqrt[4]{2}}{\sqrt{8}}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.3.1.1

Factor $4$ out of $8$ .

$\sin (x) = \pm \frac{\sqrt[4]{2}}{\sqrt{4 (2)}}$

Step 3.3.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.3.2

Pull terms out from under the radical.

$\sin (x) = \pm \frac{\sqrt[4]{2}}{2 \sqrt{2}}$

Step 3.4

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

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

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

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

$\sin (x) = \pm \frac{\sqrt[4]{2} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 3.5.2

Move $\sqrt{2}$ .

$\sin (x) = \pm \frac{\sqrt[4]{2} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.5.3

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

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

Step 3.5.4

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

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

Step 3.5.5

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

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

Step 3.5.6

Add $1$ and $1$ .

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

Step 3.5.7

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

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Step 3.5.7.1

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

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

Step 3.5.7.2

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

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

Step 3.5.7.3

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

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

Step 3.5.7.4

Cancel the common factor of $2$ .

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Step 3.5.7.4.1

Cancel the common factor.

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

Step 3.5.7.4.2

Rewrite the expression.

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

Step 3.5.7.5

Evaluate the exponent.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

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

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Step 3.6.1.1

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

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

Step 3.6.1.2

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

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

Step 3.6.1.3

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

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

Step 3.6.2

Combine using the product rule for radicals.

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

Step 3.6.3

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.6.3.1

Multiply $2$ by $2^{2}$ .

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Step 3.6.3.1.1

Raise $2$ to the power of $1$ .

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

Step 3.6.3.1.2

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

$\sin (x) = \pm \frac{\sqrt[4]{2^{1 + 2}}}{2 \cdot 2}$

Step 3.6.3.2

Add $1$ and $2$ .

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

Step 3.7

Simplify the expression.

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Step 3.7.1

Multiply $2$ by $2$ .

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

Step 3.7.2

Raise $2$ to the power of $3$ .

$\sin (x) = \pm \frac{\sqrt[4]{8}}{4}$

Step 4

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

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Step 4.1

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

$\sin (x) = \frac{\sqrt[4]{8}}{4}$

Step 4.2

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

$\sin (x) = - \frac{\sqrt[4]{8}}{4}$

Step 4.3

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

$\sin (x) = \frac{\sqrt[4]{8}}{4}, - \frac{\sqrt[4]{8}}{4}$

Step 5

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

$\sin (x) = \frac{\sqrt[4]{8}}{4}$

$\sin (x) = - \frac{\sqrt[4]{8}}{4}$

Step 6

Solve for $x$ in $\sin (x) = \frac{\sqrt[4]{8}}{4}$ .

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Step 6.1

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

$x = \arcsin (\frac{\sqrt[4]{8}}{4})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $\arcsin (\frac{\sqrt[4]{8}}{4})$ .

$x = 0.43393925$

Step 6.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.43393925$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Remove parentheses.

$x = 3.14159265 - 0.43393925$

Step 6.4.2

Remove parentheses.

$x = (3.14159265) - 0.43393925$

Step 6.4.3

Subtract $0.43393925$ from $3.14159265$ .

$x = 2.70765339$

Step 6.5

Find the period of $\sin (x)$ .

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Step 6.5.1

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

$\frac{2 π}{| b |}$

Step 6.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

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

$x = 0.43393925 + 2 π n, 2.70765339 + 2 π n$ , for any integer $n$

Step 7

Solve for $x$ in $\sin (x) = - \frac{\sqrt[4]{8}}{4}$ .

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Step 7.1

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

$x = \arcsin (- \frac{\sqrt[4]{8}}{4})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Evaluate $\arcsin (- \frac{\sqrt[4]{8}}{4})$ .

$x = - 0.43393925$

Step 7.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 0.43393925 + 3.14159265$

Step 7.4

Simplify the expression to find the second solution.

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Step 7.4.1

Subtract $2 π$ from $2 (3.14159265) + 0.43393925 + 3.14159265$ .

$x = 2 (3.14159265) + 0.43393925 + 3.14159265 - 2 π$

Step 7.4.2

The resulting angle of $3.5755319$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.43393925 + 3.14159265$ .

$x = 3.5755319$

Step 7.5

Find the period of $\sin (x)$ .

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Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

Add $2 π$ to every negative angle to get positive angles.

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Step 7.6.1

Add $2 π$ to $- 0.43393925$ to find the positive angle.

$- 0.43393925 + 2 π$

Step 7.6.2

Subtract $0.43393925$ from $2 π$ .

$5.84924605$

Step 7.6.3

List the new angles.

$x = 5.84924605$

Step 7.7

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

$x = 3.5755319 + 2 π n, 5.84924605 + 2 π n$ , for any integer $n$

Step 8

List all of the solutions.

$x = 0.43393925 + 2 π n, 2.70765339 + 2 π n, 3.5755319 + 2 π n, 5.84924605 + 2 π n$ , for any integer $n$

Step 9

Consolidate the solutions.

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Step 9.1

Consolidate $0.43393925 + 2 π n$ and $3.5755319 + 2 π n$ to $0.43393925 + π n$ .

$x = 0.43393925 + π n, 2.70765339 + 2 π n, 5.84924605 + 2 π n$ , for any integer $n$

Step 9.2

Consolidate $2.70765339 + 2 π n$ and $5.84924605 + 2 π n$ to $2.70765339 + π n$ .

$x = 0.43393925 + π n, 2.70765339 + π n$ , for any integer $n$