Trigonometry Examples

$2 \cos (120) + i \sin (120) + 4 \cos (35) + i \sin (35)$

Step 1

Simplify each term.

Tap for more steps...

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.

$2 (- \cos (60)) + i \sin (120) + 4 \cos (35) + i \sin (35)$

Step 1.2

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

$2 (- \frac{1}{2}) + i \sin (120) + 4 \cos (35) + i \sin (35)$

Step 1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.1

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

$2 (\frac{- 1}{2}) + i \sin (120) + 4 \cos (35) + i \sin (35)$

Step 1.3.2

Cancel the common factor.

$2 (\frac{- 1}{2}) + i \sin (120) + 4 \cos (35) + i \sin (35)$

Step 1.3.3

Rewrite the expression.

$- 1 + i \sin (120) + 4 \cos (35) + i \sin (35)$

Step 1.4

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

$- 1 + i \sin (60) + 4 \cos (35) + i \sin (35)$

Step 1.5

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

$- 1 + i \frac{\sqrt{3}}{2} + 4 \cos (35) + i \sin (35)$

Step 1.6

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

$- 1 + \frac{i \sqrt{3}}{2} + 4 \cos (35) + i \sin (35)$

Step 2

Reorder $- 1 + \frac{i \sqrt{3}}{2}$ and $4 \cos (35)$ .

$4 \cos (35) - 1 + \frac{i \sqrt{3}}{2} + i \sin (35)$