Algebra Examples

$3 (\cos (225) + i \sin (225))$

Step 1

Simplify each term.

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Step 1.1

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 third quadrant.

$3 (- \cos (45) + i \sin (225))$

Step 1.2

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

$3 (- \frac{\sqrt{2}}{2} + i \sin (225))$

Step 1.3

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 third quadrant.

$3 (- \frac{\sqrt{2}}{2} + i (- \sin (45)))$

Step 1.4

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

$3 (- \frac{\sqrt{2}}{2} + i (- \frac{\sqrt{2}}{2}))$

Step 1.5

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

$3 (- \frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})$

Step 2

Apply the distributive property.

$3 (- \frac{\sqrt{2}}{2}) + 3 (- \frac{i \sqrt{2}}{2})$

Step 3

Multiply $3 (- \frac{\sqrt{2}}{2})$ .

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Step 3.1

Multiply $- 1$ by $3$ .

$- 3 \frac{\sqrt{2}}{2} + 3 (- \frac{i \sqrt{2}}{2})$

Step 3.2

Combine $- 3$ and $\frac{\sqrt{2}}{2}$ .

$\frac{- 3 \sqrt{2}}{2} + 3 (- \frac{i \sqrt{2}}{2})$

Step 4

Multiply $3 (- \frac{i \sqrt{2}}{2})$ .

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Step 4.1

Multiply $- 1$ by $3$ .

$\frac{- 3 \sqrt{2}}{2} - 3 \frac{i \sqrt{2}}{2}$

Step 4.2

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

$\frac{- 3 \sqrt{2}}{2} + \frac{- 3 (i \sqrt{2})}{2}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{3 \sqrt{2}}{2} + \frac{- 3 (i \sqrt{2})}{2}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{3 \sqrt{2}}{2} - \frac{3 i \sqrt{2}}{2}$