Trigonometry Examples

$\cos (40 - 100) + i \sin (40 - 100)$

Step 1

Subtract $100$ from $40$ .

$\cos (- 60) + i \sin (40 - 100)$

Step 2

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

$\cos (60) + i \sin (40 - 100)$

Step 3

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

$\frac{1}{2} + i \sin (40 - 100)$

Step 4

Subtract $100$ from $40$ .

$\frac{1}{2} + i \sin (- 60)$

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 fourth quadrant.

$\frac{1}{2} + i (- \sin (60))$

Step 6

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

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

Step 7

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

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