Simplify sin((5pi)/12)cos(pi/3)+cos((5pi)/12)sin(pi/3)

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Trigonometry

$\sin (\frac{5 π}{12}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Simplify terms.

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Simplify each term.

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The exact value of $\sin (\frac{5 π}{12})$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\sin (\frac{π}{6} + \frac{π}{4}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Apply the sum of angles identity.

$(\sin (\frac{π}{6}) \cos (\frac{π}{4}) + \cos (\frac{π}{6}) \sin (\frac{π}{4})) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

$(\frac{1}{2} \cos (\frac{π}{4}) + \cos (\frac{π}{6}) \sin (\frac{π}{4})) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (\frac{π}{6}) \sin (\frac{π}{4})) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (\frac{π}{4})) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

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Simplify each term.

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

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

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Multiply $3$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cos (\frac{π}{3}) + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{2} + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

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

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Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $\frac{1}{2}$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 \cdot 2} + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

Multiply $4$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \cos (\frac{5 π}{12}) \sin (\frac{π}{3})$

The exact value of $\cos (\frac{5 π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sqrt{2} + \sqrt{6}}{8} + \cos (\frac{π}{6} + \frac{π}{4}) \sin (\frac{π}{3})$

Apply the sum of angles identity.

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\cos (\frac{π}{6}) \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4})) \sin (\frac{π}{3})$

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{3}}{2} \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4})) \sin (\frac{π}{3})$

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (\frac{π}{6}) \sin (\frac{π}{4})) \sin (\frac{π}{3})$

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (\frac{π}{4})) \sin (\frac{π}{3})$

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \sin (\frac{π}{3})$

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

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Simplify each term.

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

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \sin (\frac{π}{3})$

Combine using the product rule for radicals.

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \sin (\frac{π}{3})$

Multiply $3$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \sin (\frac{π}{3})$

Multiply $2$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \sin (\frac{π}{3})$

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

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \sin (\frac{π}{3})$

Multiply $2$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \sin (\frac{π}{3})$

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6} - \sqrt{2}}{4} \sin (\frac{π}{3})$

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

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{3}}{2}$

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{3}}{2}$ .

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Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{(\sqrt{6} - \sqrt{2}) \sqrt{3}}{4 \cdot 2}$

Multiply $4$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{(\sqrt{6} - \sqrt{2}) \sqrt{3}}{8}$

Apply the distributive property.

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6} \sqrt{3} - \sqrt{2} \sqrt{3}}{8}$

Combine using the product rule for radicals.

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6 \cdot 3} - \sqrt{2} \sqrt{3}}{8}$

Multiply $- \sqrt{2} \sqrt{3}$ .

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Combine using the product rule for radicals.

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6 \cdot 3} - \sqrt{3 \cdot 2}}{8}$

Multiply $3$ by $2$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{6 \cdot 3} - \sqrt{6}}{8}$

Simplify each term.

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Multiply $6$ by $3$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{18} - \sqrt{6}}{8}$

Rewrite $18$ as $3^{2} \cdot 2$ .

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Factor $9$ out of $18$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{9 (2)} - \sqrt{6}}{8}$

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{\sqrt{3^{2} \cdot 2} - \sqrt{6}}{8}$

Pull terms out from under the radical.

$\frac{\sqrt{2} + \sqrt{6}}{8} + \frac{3 \sqrt{2} - \sqrt{6}}{8}$

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6} + 3 \sqrt{2} - \sqrt{6}}{8}$

Simplify the numerator.

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

$\frac{4 \sqrt{2} + \sqrt{6} - \sqrt{6}}{8}$

Subtract $\sqrt{6}$ from $\sqrt{6}$ .

$\frac{4 \sqrt{2} + 0}{8}$

Add $4 \sqrt{2}$ and $0$ .

$\frac{4 \sqrt{2}}{8}$

Cancel the common factor of $4$ and $8$ .

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Factor $4$ out of $4 \sqrt{2}$ .

$\frac{4 (\sqrt{2})}{8}$

Cancel the common factors.

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Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{2}}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$

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Name

Name	seven hundred two million one hundred sixty-four thousand six hundred forty-four

Interesting facts

702164644 has 8 divisors, whose sum is 1579870458
The reverse of 702164644 is 446461207
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 34
Digital Root 7

Name

Name	two hundred thirty-one million two hundred sixty-nine thousand six hundred sixty-two

Interesting facts

231269662 has 16 divisors, whose sum is 379168272
The reverse of 231269662 is 266962132
Previous prime number is 67

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 37
Digital Root 1

Name

Name	eight hundred seventy-seven million nine hundred fifty-seven thousand one hundred seventy-one

Interesting facts

877957171 has 8 divisors, whose sum is 1080562784
The reverse of 877957171 is 171759778
Previous prime number is 13

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 52
Digital Root 7

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