Trigonometry Examples

$\cos (\frac{2 π}{3} + \frac{π}{4})$

Step 1

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\cos (\frac{2 π}{3} \cdot \frac{4}{4} + \frac{π}{4})$

Step 2

To write $\frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\cos (\frac{2 π}{3} \cdot \frac{4}{4} + \frac{π}{4} \cdot \frac{3}{3})$

Step 3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2 π}{3}$ by $\frac{4}{4}$ .

$\cos (\frac{2 π \cdot 4}{3 \cdot 4} + \frac{π}{4} \cdot \frac{3}{3})$

Step 3.2

Multiply $3$ by $4$ .

$\cos (\frac{2 π \cdot 4}{12} + \frac{π}{4} \cdot \frac{3}{3})$

Step 3.3

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$\cos (\frac{2 π \cdot 4}{12} + \frac{π \cdot 3}{4 \cdot 3})$

Step 3.4

Multiply $4$ by $3$ .

$\cos (\frac{2 π \cdot 4}{12} + \frac{π \cdot 3}{12})$

Step 4

Combine the numerators over the common denominator.

$\cos (\frac{2 π \cdot 4 + π \cdot 3}{12})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $4$ by $2$ .

$\cos (\frac{8 π + π \cdot 3}{12})$

Step 5.2

Move $3$ to the left of $π$ .

$\cos (\frac{8 π + 3 \cdot π}{12})$

Step 5.3

Add $8 π$ and $3 π$ .

$\cos (\frac{11 π}{12})$

Step 6

The exact value of $\cos (\frac{11 π}{12})$ is $- \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 6.1

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.

$- \cos (\frac{π}{12})$

Step 6.2

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

$- \cos (\frac{π}{4} - \frac{π}{6})$

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

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Step 6.8.1

Simplify each term.

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Step 6.8.1.1

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

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Step 6.8.1.1.1

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

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

Step 6.8.1.1.2

Combine using the product rule for radicals.

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

Step 6.8.1.1.3

Multiply $2$ by $3$ .

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

Step 6.8.1.1.4

Multiply $2$ by $2$ .

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

Step 6.8.1.2

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

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Step 6.8.1.2.1

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

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

Step 6.8.1.2.2

Multiply $2$ by $2$ .

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

Step 6.8.2

Combine the numerators over the common denominator.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$