Find the Exact Value sin(pi/3)cos(pi)-cos(pi/3)sin((3pi)/2)

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

Step 1

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

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

Step 2

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.

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

Step 3

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

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

Step 4

Multiply $- 1$ by $1$ .

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

Step 5

Combine $\frac{\sqrt{3}}{2}$ and $- 1$ .

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

Step 6

Simplify the numerator.

Move $- 1$ to the left of $\sqrt{3}$ .

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

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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 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

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

Multiply $- 1$ by $- 1$ .

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

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

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

Step 13

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.36602540 \dots$

Find the Exact Value sin(pi/3)cos(pi)-cos(pi/3)sin((3pi)/2)

Related Questions

Support LCMGCF.com by disabling your adblocker.