Calculus Examples

$(\cos (0) + \cos (\frac{π}{8}) + \cos (\frac{π}{4}) + \cos (\frac{3 π}{8})) (\frac{π}{8})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (0)$ is $1$ .

$(1 + \cos (\frac{π}{8}) + \cos (\frac{π}{4}) + \cos (\frac{3 π}{8})) \frac{π}{8}$

Step 1.2

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

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Step 1.2.1

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

$(1 + \cos (\frac{\frac{π}{4}}{2}) + \cos (\frac{π}{4}) + \cos (\frac{3 π}{8})) \frac{π}{8}$

Step 1.2.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Simplify $\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

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Step 1.2.5.1

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

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

Step 1.2.5.2

Combine the numerators over the common denominator.

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

Step 1.2.5.3

Multiply the numerator by the reciprocal of the denominator.

$(1 + \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} + \cos (\frac{π}{4}) + \cos (\frac{3 π}{8})) \frac{π}{8}$

Step 1.2.5.4

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

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Step 1.2.5.4.1

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

$(1 + \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} + \cos (\frac{π}{4}) + \cos (\frac{3 π}{8})) \frac{π}{8}$

Step 1.2.5.4.2

Multiply $2$ by $2$ .

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

Step 1.2.5.5

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

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

Step 1.2.5.6

Simplify the denominator.

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Step 1.2.5.6.1

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

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

Step 1.2.5.6.2

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

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

Step 1.3

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \cos (\frac{3 π}{8})) \frac{π}{8}$

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

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

Step 1.4.3

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

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

Step 1.4.4

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

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Step 1.4.4.1

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.

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

Step 1.4.4.2

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}) \frac{π}{8}$

Step 1.4.4.3

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}}) \frac{π}{8}$

Step 1.4.4.4

Combine the numerators over the common denominator.

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}) \frac{π}{8}$

Step 1.4.4.5

Multiply the numerator by the reciprocal of the denominator.

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}) \frac{π}{8}$

Step 1.4.4.6

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

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Step 1.4.4.6.1

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}) \frac{π}{8}$

Step 1.4.4.6.2

Multiply $2$ by $2$ .

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \sqrt{\frac{2 - \sqrt{2}}{4}}) \frac{π}{8}$

Step 1.4.4.7

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}) \frac{π}{8}$

Step 1.4.4.8

Simplify the denominator.

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Step 1.4.4.8.1

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}}) \frac{π}{8}$

Step 1.4.4.8.2

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

$(1 + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2 - \sqrt{2}}}{2}) \frac{π}{8}$

Step 2

Simplify by multiplying through.

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Step 2.1

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

$(\frac{2}{2} + \frac{\sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2 - \sqrt{2}}}{2}) \frac{π}{8}$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine the numerators over the common denominator.

$(\frac{2 + \sqrt{2 + \sqrt{2}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2 - \sqrt{2}}}{2}) \frac{π}{8}$

Step 2.2.2

Combine the numerators over the common denominator.

$(\frac{2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}}{2} + \frac{\sqrt{2}}{2}) \frac{π}{8}$

Step 2.3

Apply the distributive property.

$\frac{2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}}{2} \cdot \frac{π}{8} + \frac{\sqrt{2}}{2} \cdot \frac{π}{8}$

Step 3

Multiply $\frac{2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}}{2} \cdot \frac{π}{8}$ .

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Step 3.1

Multiply $\frac{2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}}{2}$ by $\frac{π}{8}$ .

$\frac{(2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}) π}{2 \cdot 8} + \frac{\sqrt{2}}{2} \cdot \frac{π}{8}$

Step 3.2

Multiply $2$ by $8$ .

$\frac{(2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}) π}{16} + \frac{\sqrt{2}}{2} \cdot \frac{π}{8}$

Step 4

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{π}{8}$ .

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Step 4.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{π}{8}$ .

$\frac{(2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}) π}{16} + \frac{\sqrt{2} π}{2 \cdot 8}$

Step 4.2

Multiply $2$ by $8$ .

$\frac{(2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}) π}{16} + \frac{\sqrt{2} π}{16}$

Step 5

Combine the numerators over the common denominator.

$\frac{(2 + \sqrt{2 + \sqrt{2}} + \sqrt{2 - \sqrt{2}}) π + \sqrt{2} π}{16}$

Step 6

Apply the distributive property.

$\frac{2 π + \sqrt{2 + \sqrt{2}} π + \sqrt{2 - \sqrt{2}} π + \sqrt{2} π}{16}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2 π + \sqrt{2 + \sqrt{2}} π + \sqrt{2 - \sqrt{2}} π + \sqrt{2} π}{16}$

Decimal Form:

$1.18346534 \dots$