Precalculus Examples

$5 (\cos (135 °) + i \sin (135 °))$

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 second quadrant.

$5 (- \cos (45) + i \sin (135 °))$

Step 1.2

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

$5 (- \frac{\sqrt{2}}{2} + i \sin (135 °))$

Step 1.3

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

$5 (- \frac{\sqrt{2}}{2} + i \sin (45))$

Step 1.4

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

$5 (- \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})$

Step 1.5

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

$5 (- \frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 2

Apply the distributive property.

$5 (- \frac{\sqrt{2}}{2}) + 5 \frac{i \sqrt{2}}{2}$

Step 3

Multiply $5 (- \frac{\sqrt{2}}{2})$ .

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Step 3.1

Multiply $- 1$ by $5$ .

$- 5 \frac{\sqrt{2}}{2} + 5 \frac{i \sqrt{2}}{2}$

Step 3.2

Combine $- 5$ and $\frac{\sqrt{2}}{2}$ .

$\frac{- 5 \sqrt{2}}{2} + 5 \frac{i \sqrt{2}}{2}$

Step 4

Combine $5$ and $\frac{i \sqrt{2}}{2}$ .

$\frac{- 5 \sqrt{2}}{2} + \frac{5 (i \sqrt{2})}{2}$

Step 5

Move the negative in front of the fraction.

$- \frac{5 \sqrt{2}}{2} + \frac{5 i \sqrt{2}}{2}$