Trigonometry Examples

$\cos (\frac{3 π}{4} - \frac{5 π}{6})$

Step 1

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

$\cos (\frac{3 π}{4} \cdot \frac{3}{3} - \frac{5 π}{6})$

Step 2

To write $- \frac{5 π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\cos (\frac{3 π}{4} \cdot \frac{3}{3} - \frac{5 π}{6} \cdot \frac{2}{2})$

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 $\frac{3 π}{4}$ by $\frac{3}{3}$ .

$\cos (\frac{3 π \cdot 3}{4 \cdot 3} - \frac{5 π}{6} \cdot \frac{2}{2})$

Step 3.2

Multiply $4$ by $3$ .

$\cos (\frac{3 π \cdot 3}{12} - \frac{5 π}{6} \cdot \frac{2}{2})$

Step 3.3

Multiply $\frac{5 π}{6}$ by $\frac{2}{2}$ .

$\cos (\frac{3 π \cdot 3}{12} - \frac{5 π \cdot 2}{6 \cdot 2})$

Step 3.4

Multiply $6$ by $2$ .

$\cos (\frac{3 π \cdot 3}{12} - \frac{5 π \cdot 2}{12})$

Step 4

Combine the numerators over the common denominator.

$\cos (\frac{3 π \cdot 3 - 5 π \cdot 2}{12})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $3$ by $3$ .

$\cos (\frac{9 π - 5 π \cdot 2}{12})$

Step 5.2

Multiply $2$ by $- 5$ .

$\cos (\frac{9 π - 10 π}{12})$

Step 5.3

Subtract $10 π$ from $9 π$ .

$\cos (\frac{- π}{12})$

Step 6

Move the negative in front of the fraction.

$\cos (- \frac{π}{12})$

Step 7

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (\frac{23 π}{12})$

Step 8

The exact value of $\cos (\frac{23 π}{12})$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 8.1

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

$\cos (\frac{π}{12})$

Step 8.2

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 8.3

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 8.4

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 8.5

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 8.6

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 8.7

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 8.8

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

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Step 8.8.1

Simplify each term.

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Step 8.8.1.1

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

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Step 8.8.1.1.1

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 8.8.1.1.2

Combine using the product rule for radicals.

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

Step 8.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 8.8.1.1.4

Multiply $2$ by $2$ .

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

Step 8.8.1.2

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

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Step 8.8.1.2.1

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

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

Step 8.8.1.2.2

Multiply $2$ by $2$ .

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

Step 8.8.2

Combine the numerators over the common denominator.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$