Precalculus Examples

$\sin^{2} (x) = \frac{3}{10}$

Step 1

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

$\sin (x) = \pm \sqrt{\frac{3}{10}}$

Step 2

Simplify $\pm \sqrt{\frac{3}{10}}$ .

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Step 2.1

Rewrite $\sqrt{\frac{3}{10}}$ as $\frac{\sqrt{3}}{\sqrt{10}}$ .

$\sin (x) = \pm \frac{\sqrt{3}}{\sqrt{10}}$

Step 2.2

Multiply $\frac{\sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (x) = \pm \frac{\sqrt{3}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 2.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 2.3.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 2.3.4

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

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 2.3.5

Add $1$ and $1$ .

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 2.3.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 2.3.6.2

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

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 2.3.6.3

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

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.3.6.4.2

Rewrite the expression.

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{10^{1}}$

Step 2.3.6.5

Evaluate the exponent.

$\sin (x) = \pm \frac{\sqrt{3} \sqrt{10}}{10}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Combine using the product rule for radicals.

$\sin (x) = \pm \frac{\sqrt{3 \cdot 10}}{10}$

Step 2.4.2

Multiply $3$ by $10$ .

$\sin (x) = \pm \frac{\sqrt{30}}{10}$

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.

$\sin (x) = \frac{\sqrt{30}}{10}$

Step 3.2

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

$\sin (x) = - \frac{\sqrt{30}}{10}$

Step 3.3

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

$\sin (x) = \frac{\sqrt{30}}{10}, - \frac{\sqrt{30}}{10}$

Step 4

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

$\sin (x) = \frac{\sqrt{30}}{10}$

$\sin (x) = - \frac{\sqrt{30}}{10}$

Step 5

Solve for $x$ in $\sin (x) = \frac{\sqrt{30}}{10}$ .

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Step 5.1

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

$x = \arcsin (\frac{\sqrt{30}}{10})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Evaluate $\arcsin (\frac{\sqrt{30}}{10})$ .

$x = 0.57963974$

Step 5.3

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.

$x = (3.14159265) - 0.57963974$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Remove parentheses.

$x = 3.14159265 - 0.57963974$

Step 5.4.2

Remove parentheses.

$x = (3.14159265) - 0.57963974$

Step 5.4.3

Subtract $0.57963974$ from $3.14159265$ .

$x = 2.56195291$

Step 5.5

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

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Step 5.5.1

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

$\frac{2 π}{| b |}$

Step 5.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.5.3

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

$\frac{2 π}{1}$

Step 5.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.6

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

$x = 0.57963974 + 2 π n, 2.56195291 + 2 π n$ , for any integer $n$

Step 6

Solve for $x$ in $\sin (x) = - \frac{\sqrt{30}}{10}$ .

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Step 6.1

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

$x = \arcsin (- \frac{\sqrt{30}}{10})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $\arcsin (- \frac{\sqrt{30}}{10})$ .

$x = - 0.57963974$

Step 6.3

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.

$x = 2 (3.14159265) + 0.57963974 + 3.14159265$

Step 6.4

Simplify the expression to find the second solution.

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Step 6.4.1

Subtract $2 π$ from $2 (3.14159265) + 0.57963974 + 3.14159265$ .

$x = 2 (3.14159265) + 0.57963974 + 3.14159265 - 2 π$

Step 6.4.2

The resulting angle of $3.72123239$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.57963974 + 3.14159265$ .

$x = 3.72123239$

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

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

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Step 6.6.1

Add $2 π$ to $- 0.57963974$ to find the positive angle.

$- 0.57963974 + 2 π$

Step 6.6.2

Subtract $0.57963974$ from $2 π$ .

$5.70354556$

Step 6.6.3

List the new angles.

$x = 5.70354556$

Step 6.7

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

$x = 3.72123239 + 2 π n, 5.70354556 + 2 π n$ , for any integer $n$

Step 7

List all of the solutions.

$x = 0.57963974 + 2 π n, 2.56195291 + 2 π n, 3.72123239 + 2 π n, 5.70354556 + 2 π n$ , for any integer $n$

Step 8

Consolidate the solutions.

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Step 8.1

Consolidate $0.57963974 + 2 π n$ and $3.72123239 + 2 π n$ to $0.57963974 + π n$ .

$x = 0.57963974 + π n, 2.56195291 + 2 π n, 5.70354556 + 2 π n$ , for any integer $n$

Step 8.2

Consolidate $2.56195291 + 2 π n$ and $5.70354556 + 2 π n$ to $2.56195291 + π n$ .

$x = 0.57963974 + π n, 2.56195291 + π n$ , for any integer $n$