Trigonometry Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

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.

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

Step 1.3

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

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $2$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

Tap for more steps...

Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.8.5

Add $1$ and $1$ .

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

Step 1.8.6

Multiply $2$ by $2$ .

$- \frac{1}{4} + \frac{{\sqrt{3}}^{2}}{4}$

Step 1.9

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

Tap for more steps...

Step 1.9.1

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

$- \frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{4}$

Step 1.9.2

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

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

Step 1.9.3

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

$- \frac{1}{4} + \frac{3^{\frac{2}{2}}}{4}$

Step 1.9.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.9.4.1

Cancel the common factor.

$- \frac{1}{4} + \frac{3^{\frac{2}{2}}}{4}$

Step 1.9.4.2

Rewrite the expression.

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

Step 1.9.5

Evaluate the exponent.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $- 1$ and $3$ .

$\frac{2}{4}$

Step 2.3

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

Tap for more steps...

Step 2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.1

Factor $2$ out of $4$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$