Convert to Rectangular 5(cos((7pi)/4)+isin((7pi)/4))

$5 (\cos (\frac{7 π}{4}) + i \sin (\frac{7 π}{4}))$

Step 1

Simplify each term.

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

$5 (\cos (\frac{π}{4}) + i \sin (\frac{7 π}{4}))$

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

$5 (\frac{\sqrt{2}}{2} + i \sin (\frac{7 π}{4}))$

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.

$5 (\frac{\sqrt{2}}{2} + i (- \sin (\frac{π}{4})))$

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

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

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

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

Step 2

Simplify terms.

Apply the distributive property.

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

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

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

Step 3

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

Multiply $- 1$ by $5$ .

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

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

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

Step 4

Move the negative in front of the fraction.

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

Convert to Rectangular 5(cos((7pi)/4)+isin((7pi)/4))

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