Precalculus Examples

$a = 3 (\cos (15) + i \sin (15))$ , $b = 2 (\cos (10) + i \sin (10))$

Step 1

Reduce the system.

$a = 3 (\cos (15) + i \sin (15)), b = 2 (\cos (10) + i \sin (10))$

Step 2

Simplify the right side.

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Step 2.1

Simplify $3 (\cos (15) + i \sin (15))$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 2.1.1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$a = 3 (\cos (45 - 30) + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.2

Separate negation.

$a = 3 (\cos (45 - (30)) + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.3

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

$a = 3 (\cos (45) \cos (30) + \sin (45) \sin (30) + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$a = 3 (\frac{\sqrt{2}}{2} \cdot \cos (30) + \sin (45) \sin (30) + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$a = 3 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30) + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$a = 3 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \sin (30) + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$a = 3 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8

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

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Step 2.1.1.1.8.1

Simplify each term.

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Step 2.1.1.1.8.1.1

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

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Step 2.1.1.1.8.1.1.1

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

$a = 3 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8.1.1.2

Combine using the product rule for radicals.

$a = 3 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8.1.1.3

Multiply $2$ by $3$ .

$a = 3 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8.1.1.4

Multiply $2$ by $2$ .

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

$b = 2 (\cos (10) + i \sin (10))$

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8.1.2

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

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Step 2.1.1.1.8.1.2.1

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

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8.1.2.2

Multiply $2$ by $2$ .

$a = 3 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.1.8.2

Combine the numerators over the common denominator.

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1.1.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - 30))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.2

Separate negation.

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - (30)))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.3

Apply the difference of angles identity.

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\sin (45) \cos (30) - \cos (45) \sin (30)))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \cos (30) - \cos (45) \sin (30)))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \sin (30)))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8

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

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Step 2.1.1.2.8.1

Simplify each term.

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Step 2.1.1.2.8.1.1

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

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Step 2.1.1.2.8.1.1.1

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

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8.1.1.2

Combine using the product rule for radicals.

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8.1.1.3

Multiply $2$ by $3$ .

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8.1.1.4

Multiply $2$ by $2$ .

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

$b = 2 (\cos (10) + i \sin (10))$

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8.1.2

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

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Step 2.1.1.2.8.1.2.1

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

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

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8.1.2.2

Multiply $2$ by $2$ .

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.2.8.2

Combine the numerators over the common denominator.

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6} - \sqrt{2}}{4}))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6} - \sqrt{2}}{4}))$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6} - \sqrt{2}}{4}))$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.1.3

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

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + \frac{i (\sqrt{6} - \sqrt{2})}{4})$

$b = 2 (\cos (10) + i \sin (10))$

$a = 3 (\frac{\sqrt{6} + \sqrt{2}}{4} + \frac{i (\sqrt{6} - \sqrt{2})}{4})$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.2

Combine fractions.

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Step 2.1.2.1

Combine the numerators over the common denominator.

$a = 3 (\frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4})$

$b = 2 (\cos (10) + i \sin (10))$

Step 2.1.2.2

Combine $3$ and $\frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4}$ .

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 2 (\cos (10) + i \sin (10))$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 2 (\cos (10) + i \sin (10))$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 2 (\cos (10) + i \sin (10))$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 2 (\cos (10) + i \sin (10))$

Step 3

Simplify the right side.

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Step 3.1

Simplify $2 (\cos (10) + i \sin (10))$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Evaluate $\cos (10)$ .

$b = 2 (0.98480775 + i \sin (10))$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 3.1.1.2

Evaluate $\sin (10)$ .

$b = 2 (0.98480775 + i \cdot 0.17364817)$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 3.1.1.3

Move $0.17364817$ to the left of $i$ .

$b = 2 (0.98480775 + 0.17364817 i)$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 2 (0.98480775 + 0.17364817 i)$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 3.1.2

Simplify by multiplying through.

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Step 3.1.2.1

Apply the distributive property.

$b = 2 \cdot 0.98480775 + 2 (0.17364817 i)$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 3.1.2.2

Multiply.

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Step 3.1.2.2.1

Multiply $2$ by $0.98480775$ .

$b = 1.9696155 + 2 (0.17364817 i)$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 3.1.2.2.2

Multiply $0.17364817$ by $2$ .

$b = 1.9696155 + 0.34729635 i$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 1.9696155 + 0.34729635 i$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 1.9696155 + 0.34729635 i$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 1.9696155 + 0.34729635 i$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

$b = 1.9696155 + 0.34729635 i$

$a = \frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$