Trigonometry Examples

$4 \cos (240) - i \sin (240)$

Step 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.

$4 (- \cos (60)) - i \sin (240)$

Step 2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$4 (- \frac{1}{2}) - i \sin (240)$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$4 (\frac{- 1}{2}) - i \sin (240)$

Step 3.2

Factor $2$ out of $4$ .

$2 (2) \frac{- 1}{2} - i \sin (240)$

Step 3.3

Cancel the common factor.

$2 \cdot 2 \frac{- 1}{2} - i \sin (240)$

Step 3.4

Rewrite the expression.

$2 \cdot - 1 - i \sin (240)$

Step 4

Multiply $2$ by $- 1$ .

$- 2 - i \sin (240)$

Step 5

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.

$- 2 - i (- \sin (60))$

Step 6

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 7

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

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Step 7.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 i \frac{\sqrt{3}}{2}$

Step 7.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

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

Step 7.3

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

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