Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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.

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $2$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

$- \frac{\sqrt{3}}{4} - \frac{1}{2} \cdot \frac{\sqrt{3}}{2}$

Step 1.8

Multiply $- \frac{1}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.8.1

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

$- \frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{2 \cdot 2}$

Step 1.8.2

Multiply $2$ by $2$ .

$- \frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{4}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{- \sqrt{3} - \sqrt{3}}{4}$

Step 2.2

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$\frac{- 2 \sqrt{3}}{4}$

Step 2.3

Cancel the common factor of $- 2$ and $4$ .

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Step 2.3.1

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $4$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 2.4

Move the negative in front of the fraction.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.86602540 \dots$