Precalculus Examples

$\frac{5}{2} \cdot (\cos (150) + i \sin (150))$

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.

$\frac{5}{2} \cdot (- \cos (30) + i \sin (150))$

Step 1.2

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

$\frac{5}{2} \cdot (- \frac{\sqrt{3}}{2} + i \sin (150))$

Step 1.3

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

$\frac{5}{2} \cdot (- \frac{\sqrt{3}}{2} + i \sin (30))$

Step 1.4

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

$\frac{5}{2} \cdot (- \frac{\sqrt{3}}{2} + i \frac{1}{2})$

Step 1.5

Combine $i$ and $\frac{1}{2}$ .

$\frac{5}{2} \cdot (- \frac{\sqrt{3}}{2} + \frac{i}{2})$

Step 2

Apply the distributive property.

$\frac{5}{2} (- \frac{\sqrt{3}}{2}) + \frac{5}{2} \cdot \frac{i}{2}$

Step 3

Multiply $\frac{5}{2} (- \frac{\sqrt{3}}{2})$ .

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Step 3.1

Multiply $\frac{5}{2}$ by $\frac{\sqrt{3}}{2}$ .

$- \frac{5 \sqrt{3}}{2 \cdot 2} + \frac{5}{2} \cdot \frac{i}{2}$

Step 3.2

Multiply $2$ by $2$ .

$- \frac{5 \sqrt{3}}{4} + \frac{5}{2} \cdot \frac{i}{2}$

Step 4

Multiply $\frac{5}{2} \cdot \frac{i}{2}$ .

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Step 4.1

Multiply $\frac{5}{2}$ by $\frac{i}{2}$ .

$- \frac{5 \sqrt{3}}{4} + \frac{5 i}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$- \frac{5 \sqrt{3}}{4} + \frac{5 i}{4}$