Trigonometry Examples

$\sin (\frac{13 π}{12}) \cos (\frac{π}{12})$

Step 1

The exact value of $\sin (\frac{13 π}{12})$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 1.1

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.

$- \sin (\frac{π}{12}) \cos (\frac{π}{12})$

Step 1.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$- \sin (\frac{π}{4} - \frac{π}{6}) \cos (\frac{π}{12})$

Step 1.3

Apply the difference of angles identity.

$- (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.5

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.6

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \cos (\frac{π}{12})$

Step 1.7

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.8

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

Tap for more steps...

Step 1.8.1

Simplify each term.

Tap for more steps...

Step 1.8.1.1

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

Tap for more steps...

Step 1.8.1.1.1

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

$- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.8.1.1.2

Combine using the product rule for radicals.

$- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.8.1.1.3

Multiply $2$ by $3$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.8.1.1.4

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{π}{12})$

Step 1.8.1.2

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

Tap for more steps...

Step 1.8.1.2.1

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

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (\frac{π}{12})$

Step 1.8.1.2.2

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (\frac{π}{12})$

Step 1.8.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} - \sqrt{2}}{4} \cos (\frac{π}{12})$

Step 2

The exact value of $\cos (\frac{π}{12})$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Tap for more steps...

Step 2.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

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

Step 2.2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

Step 2.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 2.4

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

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 2.5

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (\frac{π}{6}))$

Step 2.6

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

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 2.7

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.7.1

Simplify each term.

Tap for more steps...

Step 2.7.1.1

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

Tap for more steps...

Step 2.7.1.1.1

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

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 2.7.1.1.2

Combine using the product rule for radicals.

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 2.7.1.1.3

Multiply $2$ by $3$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 2.7.1.1.4

Multiply $2$ by $2$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 2.7.1.2

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

Tap for more steps...

Step 2.7.1.2.1

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

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2})$

Step 2.7.1.2.2

Multiply $2$ by $2$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

Step 2.7.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$

Step 3

Multiply $- \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$ .

Tap for more steps...

Step 3.1

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4}$ by $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

$- \frac{(\sqrt{6} + \sqrt{2}) (\sqrt{6} - \sqrt{2})}{4 \cdot 4}$

Step 3.2

Multiply $4$ by $4$ .

$- \frac{(\sqrt{6} + \sqrt{2}) (\sqrt{6} - \sqrt{2})}{16}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Expand $(\sqrt{6} + \sqrt{2}) (\sqrt{6} - \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$- \frac{\sqrt{6} (\sqrt{6} - \sqrt{2}) + \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Step 4.1.2

Apply the distributive property.

$- \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Step 4.1.3

Apply the distributive property.

$- \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2

Simplify and combine like terms.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Combine using the product rule for radicals.

$- \frac{\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.2

Multiply $6$ by $6$ .

$- \frac{\sqrt{36} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.3

Rewrite $36$ as $6^{2}$ .

$- \frac{\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{6 + \sqrt{6} (- \sqrt{2}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.5

Multiply $\sqrt{6} (- \sqrt{2})$ .

Tap for more steps...

Step 4.2.1.5.1

Combine using the product rule for radicals.

$- \frac{6 - \sqrt{6 \cdot 2} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.5.2

Multiply $6$ by $2$ .

$- \frac{6 - \sqrt{12} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.6

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 4.2.1.6.1

Factor $4$ out of $12$ .

$- \frac{6 - \sqrt{4 (3)} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.6.2

Rewrite $4$ as $2^{2}$ .

$- \frac{6 - \sqrt{2^{2} \cdot 3} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.7

Pull terms out from under the radical.

$- \frac{6 - (2 \sqrt{3}) + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.8

Multiply $2$ by $- 1$ .

$- \frac{6 - 2 \sqrt{3} + \sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.9

Combine using the product rule for radicals.

$- \frac{6 - 2 \sqrt{3} + \sqrt{2 \cdot 6} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.10

Multiply $2$ by $6$ .

$- \frac{6 - 2 \sqrt{3} + \sqrt{12} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.11

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 4.2.1.11.1

Factor $4$ out of $12$ .

$- \frac{6 - 2 \sqrt{3} + \sqrt{4 (3)} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.11.2

Rewrite $4$ as $2^{2}$ .

$- \frac{6 - 2 \sqrt{3} + \sqrt{2^{2} \cdot 3} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.12

Pull terms out from under the radical.

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{2} (- \sqrt{2})}{16}$

Step 4.2.1.13

Multiply $\sqrt{2} (- \sqrt{2})$ .

Tap for more steps...

Step 4.2.1.13.1

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - ({\sqrt{2}}^{1} \sqrt{2})}{16}$

Step 4.2.1.13.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{16}$

Step 4.2.1.13.3

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

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{2}}^{1 + 1}}{16}$

Step 4.2.1.13.4

Add $1$ and $1$ .

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{2}}^{2}}{16}$

Step 4.2.1.14

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

Tap for more steps...

Step 4.2.1.14.1

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

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{16}$

Step 4.2.1.14.2

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

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{16}$

Step 4.2.1.14.3

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

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{\frac{2}{2}}}{16}$

Step 4.2.1.14.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.14.4.1

Cancel the common factor.

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{\frac{2}{2}}}{16}$

Step 4.2.1.14.4.2

Rewrite the expression.

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2^{1}}{16}$

Step 4.2.1.14.5

Evaluate the exponent.

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 1 \cdot 2}{16}$

Step 4.2.1.15

Multiply $- 1$ by $2$ .

$- \frac{6 - 2 \sqrt{3} + 2 \sqrt{3} - 2}{16}$

Step 4.2.2

Subtract $2$ from $6$ .

$- \frac{4 - 2 \sqrt{3} + 2 \sqrt{3}}{16}$

Step 4.2.3

Add $- 2 \sqrt{3}$ and $2 \sqrt{3}$ .

$- \frac{4 + 0}{16}$

Step 4.2.4

Add $4$ and $0$ .

$- \frac{4}{16}$

Step 5

Cancel the common factor of $4$ and $16$ .

Tap for more steps...

Step 5.1

Factor $4$ out of $4$ .

$- \frac{4 (1)}{16}$

Step 5.2

Cancel the common factors.

Tap for more steps...

Step 5.2.1

Factor $4$ out of $16$ .

$- \frac{4 \cdot 1}{4 \cdot 4}$

Step 5.2.2

Cancel the common factor.

$- \frac{4 \cdot 1}{4 \cdot 4}$

Step 5.2.3

Rewrite the expression.

$- \frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$