Algebra Examples

$4 (\cos (\frac{9 π}{4}) + i \sin (\frac{9 π}{4})) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 1

Simplify the numerator.

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Step 1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$4 (\cos (\frac{π}{4}) + i \sin (\frac{9 π}{4})) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 1.2

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

$4 (\frac{\sqrt{2}}{2} + i \sin (\frac{9 π}{4})) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 1.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$4 (\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4})) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 1.4

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

$4 (\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 1.5

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$4 (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2}) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 1.6

Combine the numerators over the common denominator.

$4 \frac{\sqrt{2} + i \sqrt{2}}{2} \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 2

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

$\frac{4 (\sqrt{2} + i \sqrt{2})}{2} \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Reduce the expression $\frac{4 (\sqrt{2} + i \sqrt{2})}{2}$ by cancelling the common factors.

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Step 3.1.1

Factor $2$ out of $4 (\sqrt{2} + i \sqrt{2})$ .

$\frac{2 (2 (\sqrt{2} + i \sqrt{2}))}{2} \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 3.1.2

Factor $2$ out of $2$ .

$\frac{2 (2 (\sqrt{2} + i \sqrt{2}))}{2 (1)} \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 3.1.3

Cancel the common factor.

$\frac{2 (2 (\sqrt{2} + i \sqrt{2}))}{2 \cdot 1} \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 3.1.4

Rewrite the expression.

$\frac{2 (\sqrt{2} + i \sqrt{2})}{1} \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 3.2

Divide $2 (\sqrt{2} + i \sqrt{2})$ by $1$ .

$2 (\sqrt{2} + i \sqrt{2}) \div 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 4

Rewrite the division as a fraction.

$\frac{2 (\sqrt{2} + i \sqrt{2})}{2} (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$\frac{2 (\sqrt{2} + i \sqrt{2})}{2} (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 5.2

Divide $\sqrt{2} + i \sqrt{2}$ by $1$ .

$(\sqrt{2} + i \sqrt{2}) (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 6

Simplify each term.

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Step 6.1

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

$(\sqrt{2} + i \sqrt{2}) (\cos (\frac{π}{2}) + i \sin (\frac{3 π}{2}))$

Step 6.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$(\sqrt{2} + i \sqrt{2}) (0 + i \sin (\frac{3 π}{2}))$

Step 6.3

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.

$(\sqrt{2} + i \sqrt{2}) (0 + i (- \sin (\frac{π}{2})))$

Step 6.4

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

$(\sqrt{2} + i \sqrt{2}) (0 + i (- 1 \cdot 1))$

Step 6.5

Multiply $- 1$ by $1$ .

$(\sqrt{2} + i \sqrt{2}) (0 + i \cdot - 1)$

Step 6.6

Move $- 1$ to the left of $i$ .

$(\sqrt{2} + i \sqrt{2}) (0 - 1 \cdot i)$

Step 6.7

Rewrite $- 1 i$ as $- i$ .

$(\sqrt{2} + i \sqrt{2}) (0 - i)$

Step 7

Simplify by multiplying through.

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Step 7.1

Subtract $i$ from $0$ .

$(\sqrt{2} + i \sqrt{2}) (- i)$

Step 7.2

Apply the distributive property.

$\sqrt{2} (- i) + i \sqrt{2} (- i)$

Step 8

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

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Step 8.1

Raise $i$ to the power of $1$ .

$\sqrt{2} (- i) + \sqrt{2} (- (i^{1} i))$

Step 8.2

Raise $i$ to the power of $1$ .

$\sqrt{2} (- i) + \sqrt{2} (- (i^{1} i^{1}))$

Step 8.3

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

$\sqrt{2} (- i) + \sqrt{2} (- i^{1 + 1})$

Step 8.4

Add $1$ and $1$ .

$\sqrt{2} (- i) + \sqrt{2} (- i^{2})$

Step 9

Simplify each term.

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Step 9.1

Rewrite $i^{2}$ as $- 1$ .

$\sqrt{2} (- i) + \sqrt{2} (- - 1)$

Step 9.2

Multiply $- 1$ by $- 1$ .

$\sqrt{2} (- i) + \sqrt{2} \cdot 1$

Step 9.3

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

$\sqrt{2} (- i) + \sqrt{2}$

Step 10

Reorder $\sqrt{2} (- i)$ and $\sqrt{2}$ .

$\sqrt{2} + \sqrt{2} (- i)$