Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\sqrt{2} \cos (- \frac{π}{12})$

Step 4

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

$\sqrt{2} \cos (\frac{23 π}{12})$

Step 5

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

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Step 5.1

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

$\sqrt{2} \cos (\frac{π}{12})$

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

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

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Step 5.8.1

Simplify each term.

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Step 5.8.1.1

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

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Step 5.8.1.1.1

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

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

Step 5.8.1.1.2

Combine using the product rule for radicals.

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

Step 5.8.1.1.3

Multiply $2$ by $3$ .

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

Step 5.8.1.1.4

Multiply $2$ by $2$ .

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

Step 5.8.1.2

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

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Step 5.8.1.2.1

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

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

Step 5.8.1.2.2

Multiply $2$ by $2$ .

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

Step 5.8.2

Combine the numerators over the common denominator.

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

Step 6

Simplify terms.

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Step 6.1

Combine $\sqrt{2}$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Combine using the product rule for radicals.

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

Step 6.4

Combine using the product rule for radicals.

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

Step 7

Simplify each term.

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Step 7.1

Multiply $2$ by $6$ .

$\frac{\sqrt{12} + \sqrt{2 \cdot 2}}{4}$

Step 7.2

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

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Step 7.2.1

Factor $4$ out of $12$ .

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

Step 7.2.2

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

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

Step 7.3

Pull terms out from under the radical.

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

Step 7.4

Multiply $2$ by $2$ .

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

Step 7.5

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

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

Step 7.6

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

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

Step 8

Cancel the common factor of $2 \sqrt{3} + 2$ and $4$ .

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Step 8.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 8.2

Factor $2$ out of $2$ .

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

Step 8.3

Factor $2$ out of $2 (\sqrt{3}) + 2 (1)$ .

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

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $2$ out of $4$ .

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.36602540 \dots$