Calculus Examples

$\sin (x) = \frac{- 2 + \sqrt{20}}{4 \sqrt{2}}$

Step 1

Simplify $\frac{- 2 + \sqrt{20}}{4 \sqrt{2}}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 1.1.1.1

Factor $4$ out of $20$ .

$\sin (x) = \frac{- 2 + \sqrt{4 (5)}}{4 \sqrt{2}}$

Step 1.1.1.2

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

$\sin (x) = \frac{- 2 + \sqrt{2^{2} \cdot 5}}{4 \sqrt{2}}$

Step 1.1.2

Pull terms out from under the radical.

$\sin (x) = \frac{- 2 + 2 \sqrt{5}}{4 \sqrt{2}}$

Step 1.2

Cancel the common factor of $- 2 + 2 \sqrt{5}$ and $4$ .

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Step 1.2.1

Factor $2$ out of $- 2$ .

$\sin (x) = \frac{2 \cdot - 1 + 2 \sqrt{5}}{4 \sqrt{2}}$

Step 1.2.2

Factor $2$ out of $2 \sqrt{5}$ .

$\sin (x) = \frac{2 \cdot - 1 + 2 (\sqrt{5})}{4 \sqrt{2}}$

Step 1.2.3

Factor $2$ out of $2 \cdot - 1 + 2 (\sqrt{5})$ .

$\sin (x) = \frac{2 \cdot (- 1 + \sqrt{5})}{4 \sqrt{2}}$

Step 1.2.4

Cancel the common factors.

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Step 1.2.4.1

Factor $2$ out of $4 \sqrt{2}$ .

$\sin (x) = \frac{2 \cdot (- 1 + \sqrt{5})}{2 (2 \sqrt{2})}$

Step 1.2.4.2

Cancel the common factor.

$\sin (x) = \frac{2 \cdot (- 1 + \sqrt{5})}{2 (2 \sqrt{2})}$

Step 1.2.4.3

Rewrite the expression.

$\sin (x) = \frac{- 1 + \sqrt{5}}{2 \sqrt{2}}$

Step 1.3

Multiply $\frac{- 1 + \sqrt{5}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \frac{- 1 + \sqrt{5}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{- 1 + \sqrt{5}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 1.4.2

Move $\sqrt{2}$ .

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 1.4.3

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

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 1.4.4

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

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 1.4.5

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

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 1.4.6

Add $1$ and $1$ .

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 1.4.7

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

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Step 1.4.7.1

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

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 1.4.7.2

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

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 1.4.7.3

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

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 1.4.7.4

Cancel the common factor of $2$ .

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Step 1.4.7.4.1

Cancel the common factor.

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 1.4.7.4.2

Rewrite the expression.

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 \cdot 2^{1}}$

Step 1.4.7.5

Evaluate the exponent.

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{2 \cdot 2}$

Step 1.5

Multiply $2$ by $2$ .

$\sin (x) = \frac{(- 1 + \sqrt{5}) \sqrt{2}}{4}$

Step 1.6

Apply the distributive property.

$\sin (x) = \frac{- 1 \sqrt{2} + \sqrt{5} \sqrt{2}}{4}$

Step 1.7

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

$\sin (x) = \frac{- \sqrt{2} + \sqrt{5} \sqrt{2}}{4}$

Step 1.8

Combine using the product rule for radicals.

$\sin (x) = \frac{- \sqrt{2} + \sqrt{5 \cdot 2}}{4}$

Step 1.9

Multiply $5$ by $2$ .

$\sin (x) = \frac{- \sqrt{2} + \sqrt{10}}{4}$

Step 1.10

Factor $- 1$ out of $- \sqrt{2}$ .

$\sin (x) = \frac{- (\sqrt{2}) + \sqrt{10}}{4}$

Step 1.11

Factor $- 1$ out of $\sqrt{10}$ .

$\sin (x) = \frac{- (\sqrt{2}) - 1 (- \sqrt{10})}{4}$

Step 1.12

Factor $- 1$ out of $- (\sqrt{2}) - 1 (- \sqrt{10})$ .

$\sin (x) = \frac{- (\sqrt{2} - \sqrt{10})}{4}$

Step 1.13

Simplify the expression.

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Step 1.13.1

Rewrite $- (\sqrt{2} - \sqrt{10})$ as $- 1 (\sqrt{2} - \sqrt{10})$ .

$\sin (x) = \frac{- 1 (\sqrt{2} - \sqrt{10})}{4}$

Step 1.13.2

Move the negative in front of the fraction.

$\sin (x) = - \frac{\sqrt{2} - \sqrt{10}}{4}$

Step 2

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

$x = \arcsin (- \frac{\sqrt{2} - \sqrt{10}}{4})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (- \frac{\sqrt{2} - \sqrt{10}}{4})$ .

$x = 0.45227844$

Step 4

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.

$x = 2 (3.14159265) - 0.45227844 + 3.14159265$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $2 π$ from $2 (3.14159265) - 0.45227844 + 3.14159265$ .

$x = 2 (3.14159265) - 0.45227844 + 3.14159265 - 2 π$

Step 5.2

The resulting angle of $2.6893142$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) - 0.45227844 + 3.14159265$ .

$x = 2.6893142$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$x = 0.45227844 + 2 π n, 2.6893142 + 2 π n$ , for any integer $n$