Trigonometry Examples

$\sin (\frac{4 π}{3}) \cos (2 π) - \cos (\frac{4 π}{3}) \sin (\frac{3 π}{2})$

Step 1

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.

$- \sin (\frac{π}{3}) \cos (2 π) - \cos (\frac{4 π}{3}) \sin (\frac{3 π}{2})$

Step 2

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

$- \frac{\sqrt{3}}{2} \cos (2 π) - \cos (\frac{4 π}{3}) \sin (\frac{3 π}{2})$

Step 3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \frac{\sqrt{3}}{2} \cos (0) - \cos (\frac{4 π}{3}) \sin (\frac{3 π}{2})$

Step 4

The exact value of $\cos (0)$ is $1$ .

$- \frac{\sqrt{3}}{2} \cdot 1 - \cos (\frac{4 π}{3}) \sin (\frac{3 π}{2})$

Step 5

Multiply $- 1$ by $1$ .

$- \frac{\sqrt{3}}{2} - \cos (\frac{4 π}{3}) \sin (\frac{3 π}{2})$

Step 6

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.

$- \frac{\sqrt{3}}{2} - - \cos (\frac{π}{3}) \sin (\frac{3 π}{2})$

Step 7

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \frac{\sqrt{3}}{2} - - \frac{1}{2} \sin (\frac{3 π}{2})$

Step 8

Multiply $- - \frac{1}{2}$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

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

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

Step 9

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.

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

Step 10

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

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

Step 11

Multiply $- 1$ by $1$ .

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

Step 12

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

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

Step 13

Move the negative in front of the fraction.

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

Step 14

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.36602540 \dots$