Calculus Examples

$15 (\cos (\frac{23 π}{12}) + i \sin (\frac{23 π}{12}))$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{23 π}{12})$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$15 (\cos (\frac{π}{12}) + i \sin (\frac{23 π}{12}))$

Step 1.1.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$15 (\cos (\frac{π}{4} - \frac{π}{6}) + i \sin (\frac{23 π}{12}))$

Step 1.1.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$15 (\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6}) + i \sin (\frac{23 π}{12}))$

Step 1.1.4

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

$15 (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6}) + i \sin (\frac{23 π}{12}))$

Step 1.1.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$15 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6}) + i \sin (\frac{23 π}{12}))$

Step 1.1.6

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

$15 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (\frac{π}{6}) + i \sin (\frac{23 π}{12}))$

Step 1.1.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$15 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (\frac{23 π}{12}))$

Step 1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$15 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (\frac{23 π}{12}))$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

$15 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (\frac{23 π}{12}))$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

$15 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (\frac{23 π}{12}))$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

$15 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (\frac{23 π}{12}))$

Step 1.1.8.1.2

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

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Step 1.1.8.1.2.1

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

$15 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2} + i \sin (\frac{23 π}{12}))$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

$15 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (\frac{23 π}{12}))$

Step 1.1.8.2

Combine the numerators over the common denominator.

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (\frac{23 π}{12}))$

Step 1.2

The exact value of $\sin (\frac{23 π}{12})$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.2.1

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

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- \sin (\frac{π}{12})))$

Step 1.2.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- \sin (\frac{π}{4} - \frac{π}{6})))$

Step 1.2.3

Apply the difference of angles identity.

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6}))))$

Step 1.2.4

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

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6}))))$

Step 1.2.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6}))))$

Step 1.2.6

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

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6}))))$

Step 1.2.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 1.2.8

Simplify $- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$ .

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.2.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

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

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})))$

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})))$

Step 1.2.8.2

Combine the numerators over the common denominator.

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (- \frac{\sqrt{6} - \sqrt{2}}{4}))$

Step 1.3

Combine $i$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

$15 (\frac{\sqrt{6} + \sqrt{2}}{4} - \frac{i (\sqrt{6} - \sqrt{2})}{4})$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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Step 2.1.1

Apply the distributive property.

$15 \frac{\sqrt{6} + \sqrt{2} - i \sqrt{6} - i (- \sqrt{2})}{4}$

Step 2.1.2

Multiply $- i (- \sqrt{2})$ .

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Step 2.1.2.1

Multiply $- 1$ by $- 1$ .

$15 \frac{\sqrt{6} + \sqrt{2} - i \sqrt{6} + 1 i \sqrt{2}}{4}$

Step 2.1.2.2

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

$15 \frac{\sqrt{6} + \sqrt{2} - i \sqrt{6} + \sqrt{2} i}{4}$

Step 2.2

Combine $15$ and $\frac{\sqrt{6} + \sqrt{2} - i \sqrt{6} + \sqrt{2} i}{4}$ .

$\frac{15 (\sqrt{6} + \sqrt{2} - i \sqrt{6} + \sqrt{2} i)}{4}$