Calculus Examples

$\sin^{2} (0) + \sin^{2} (\frac{π}{8}) + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (0)$ is $0$ .

$0^{2} + \sin^{2} (\frac{π}{8}) + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.2

Raising $0$ to any positive power yields $0$ .

$0 + \sin^{2} (\frac{π}{8}) + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3

The exact value of $\sin (\frac{π}{8})$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.3.1

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$0 + \sin^{2} (\frac{\frac{π}{4}}{2}) + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.2

Apply the sine half-angle identity.

$0 + {(\pm \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}})}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.3

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

$0 + {\sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4

Simplify $\sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}$ .

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Step 1.3.4.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$0 + {\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.2

Write $1$ as a fraction with a common denominator.

$0 + {\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.3

Combine the numerators over the common denominator.

$0 + {\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.4

Multiply the numerator by the reciprocal of the denominator.

$0 + {\sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.5

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

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Step 1.3.4.5.1

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

$0 + {\sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.5.2

Multiply $2$ by $2$ .

$0 + {\sqrt{\frac{2 - \sqrt{2}}{4}}}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.6

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$0 + {(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}})}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.7

Simplify the denominator.

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Step 1.3.4.7.1

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

$0 + {(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}})}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.3.4.7.2

Pull terms out from under the radical, assuming positive real numbers.

$0 + {(\frac{\sqrt{2 - \sqrt{2}}}{2})}^{2} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.4

Apply the product rule to $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

$0 + \frac{{\sqrt{2 - \sqrt{2}}}^{2}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.5

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

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Step 1.5.1

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

$0 + \frac{{({(2 - \sqrt{2})}^{\frac{1}{2}})}^{2}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.5.2

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

$0 + \frac{{(2 - \sqrt{2})}^{\frac{1}{2} \cdot 2}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.5.3

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

$0 + \frac{{(2 - \sqrt{2})}^{\frac{2}{2}}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$0 + \frac{{(2 - \sqrt{2})}^{\frac{2}{2}}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.5.4.2

Rewrite the expression.

$0 + \frac{{(2 - \sqrt{2})}^{1}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.5.5

Simplify.

$0 + \frac{2 - \sqrt{2}}{2^{2}} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.6

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

$0 + \frac{2 - \sqrt{2}}{4} + \sin^{2} (\frac{π}{4}) + \sin^{2} (\frac{3 π}{8})$

Step 1.7

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$0 + \frac{2 - \sqrt{2}}{4} + {(\frac{\sqrt{2}}{2})}^{2} + \sin^{2} (\frac{3 π}{8})$

Step 1.8

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{{\sqrt{2}}^{2}}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.9

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

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Step 1.9.1

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.9.2

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.9.3

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2^{\frac{2}{2}}}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2^{\frac{2}{2}}}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.9.4.2

Rewrite the expression.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2^{1}}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.9.5

Evaluate the exponent.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2}{2^{2}} + \sin^{2} (\frac{3 π}{8})$

Step 1.10

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2}{4} + \sin^{2} (\frac{3 π}{8})$

Step 1.11

Cancel the common factor of $2$ and $4$ .

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Step 1.11.1

Factor $2$ out of $2$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2 (1)}{4} + \sin^{2} (\frac{3 π}{8})$

Step 1.11.2

Cancel the common factors.

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Step 1.11.2.1

Factor $2$ out of $4$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2 \cdot 1}{2 \cdot 2} + \sin^{2} (\frac{3 π}{8})$

Step 1.11.2.2

Cancel the common factor.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{2 \cdot 1}{2 \cdot 2} + \sin^{2} (\frac{3 π}{8})$

Step 1.11.2.3

Rewrite the expression.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \sin^{2} (\frac{3 π}{8})$

Step 1.12

The exact value of $\sin (\frac{3 π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.12.1

Rewrite $\frac{3 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \sin^{2} (\frac{\frac{3 π}{4}}{2})$

Step 1.12.2

Apply the sine half-angle identity.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {(\pm \sqrt{\frac{1 - \cos (\frac{3 π}{4})}{2}})}^{2}$

Step 1.12.3

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{1 - \cos (\frac{3 π}{4})}{2}}}^{2}$

Step 1.12.4

Simplify $\sqrt{\frac{1 - \cos (\frac{3 π}{4})}{2}}$ .

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Step 1.12.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{1 - - \cos (\frac{π}{4})}{2}}}^{2}$

Step 1.12.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.12.4.3

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 1.12.4.3.1

Multiply $- 1$ by $- 1$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.12.4.3.2

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.12.4.4

Write $1$ as a fraction with a common denominator.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.12.4.5

Combine the numerators over the common denominator.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}}^{2}$

Step 1.12.4.6

Multiply the numerator by the reciprocal of the denominator.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}}}^{2}$

Step 1.12.4.7

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

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Step 1.12.4.7.1

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}}}^{2}$

Step 1.12.4.7.2

Multiply $2$ by $2$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {\sqrt{\frac{2 + \sqrt{2}}{4}}}^{2}$

Step 1.12.4.8

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {(\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})}^{2}$

Step 1.12.4.9

Simplify the denominator.

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Step 1.12.4.9.1

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {(\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}})}^{2}$

Step 1.12.4.9.2

Pull terms out from under the radical, assuming positive real numbers.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + {(\frac{\sqrt{2 + \sqrt{2}}}{2})}^{2}$

Step 1.13

Apply the product rule to $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{{\sqrt{2 + \sqrt{2}}}^{2}}{2^{2}}$

Step 1.14

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

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Step 1.14.1

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{{({(2 + \sqrt{2})}^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 1.14.2

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{{(2 + \sqrt{2})}^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 1.14.3

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{{(2 + \sqrt{2})}^{\frac{2}{2}}}{2^{2}}$

Step 1.14.4

Cancel the common factor of $2$ .

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Step 1.14.4.1

Cancel the common factor.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{{(2 + \sqrt{2})}^{\frac{2}{2}}}{2^{2}}$

Step 1.14.4.2

Rewrite the expression.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{{(2 + \sqrt{2})}^{1}}{2^{2}}$

Step 1.14.5

Simplify.

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{2 + \sqrt{2}}{2^{2}}$

Step 1.15

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

$0 + \frac{2 - \sqrt{2}}{4} + \frac{1}{2} + \frac{2 + \sqrt{2}}{4}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{2 - \sqrt{2} + 2 + \sqrt{2}}{4} + \frac{1}{2}$

Step 2.2

Add $2$ and $2$ .

$\frac{4 - \sqrt{2} + \sqrt{2}}{4} + \frac{1}{2}$

Step 2.3

Add $- \sqrt{2}$ and $\sqrt{2}$ .

$\frac{4 + 0}{4} + \frac{1}{2}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $4$ and $0$ .

$\frac{4}{4} + \frac{1}{2}$

Step 2.4.2

Divide $4$ by $4$ .

$1 + \frac{1}{2}$

Step 2.4.3

Write $1$ as a fraction with a common denominator.

$\frac{2}{2} + \frac{1}{2}$

Step 2.4.4

Combine the numerators over the common denominator.

$\frac{2 + 1}{2}$

Step 2.4.5

Add $2$ and $1$ .

$\frac{3}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$