Trigonometry Examples

$\sin (\frac{2 π}{3}) \cos (\frac{π}{6}) - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1

Simplify each term.

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Step 1.1

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

$\sin (\frac{π}{3}) \cos (\frac{π}{6}) - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.2

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

$\frac{\sqrt{3}}{2} \cos (\frac{π}{6}) - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.4

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

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Step 1.4.1

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

$\frac{\sqrt{3} \sqrt{3}}{2 \cdot 2} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{2 \cdot 2} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2 \cdot 2} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sqrt{3}}^{1 + 1}}{2 \cdot 2} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.4.5

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{2 \cdot 2} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.4.6

Multiply $2$ by $2$ .

$\frac{{\sqrt{3}}^{2}}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3^{\frac{1}{2} \cdot 2}}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.5.3

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

$\frac{3^{\frac{2}{2}}}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.5.4.2

Rewrite the expression.

$\frac{3^{1}}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.5.5

Evaluate the exponent.

$\frac{3}{4} - \cos (\frac{2 π}{3}) \sin (\frac{π}{6})$

Step 1.6

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 second quadrant.

$\frac{3}{4} - - \cos (\frac{π}{3}) \sin (\frac{π}{6})$

Step 1.7

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

$\frac{3}{4} - - \frac{1}{2} \sin (\frac{π}{6})$

Step 1.8

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

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Step 1.8.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{4} + 1 (\frac{1}{2}) \sin (\frac{π}{6})$

Step 1.8.2

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

$\frac{3}{4} + \frac{1}{2} \sin (\frac{π}{6})$

Step 1.9

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{3}{4} + \frac{1}{2} \cdot \frac{1}{2}$

Step 1.10

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

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Step 1.10.1

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

$\frac{3}{4} + \frac{1}{2 \cdot 2}$

Step 1.10.2

Multiply $2$ by $2$ .

$\frac{3}{4} + \frac{1}{4}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{3 + 1}{4}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $3$ and $1$ .

$\frac{4}{4}$

Step 2.2.2

Divide $4$ by $4$ .

$1$