Find the Exact Value sin(pi/4)cos((5pi)/12)+cos(pi/4)sin((5pi)/12)

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

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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

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

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

Multiply $2$ by $4$ .

$\frac{\sqrt{2} (\sqrt{6} - \sqrt{2})}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{6} + \sqrt{2} (- \sqrt{2})}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 6} + \sqrt{2} (- \sqrt{2})}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

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

$\frac{\sqrt{2 \cdot 6} - ({\sqrt{2}}^{1} \sqrt{2})}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

$\frac{\sqrt{2 \cdot 6} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

$\frac{\sqrt{2 \cdot 6} - {\sqrt{2}}^{1 + 1}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Add $1$ and $1$ .

$\frac{\sqrt{2 \cdot 6} - {\sqrt{2}}^{2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Simplify each term.

Multiply $2$ by $6$ .

$\frac{\sqrt{12} - {\sqrt{2}}^{2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} - {\sqrt{2}}^{2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

$\frac{\sqrt{2^{2} \cdot 3} - {\sqrt{2}}^{2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Pull terms out from under the radical.

$\frac{2 \sqrt{3} - {\sqrt{2}}^{2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

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

$\frac{2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

$\frac{2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Rewrite the expression.

$\frac{2 \sqrt{3} - 2^{1}}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Evaluate the exponent.

$\frac{2 \sqrt{3} - 1 \cdot 2}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{3} - 2}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Cancel the common factor of $2 \sqrt{3} - 2$ and $8$ .

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

$\frac{2 (\sqrt{3}) - 2}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{3}) + 2 (- 1)}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

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

$\frac{2 (\sqrt{3} - 1)}{8} + \cos (\frac{π}{4}) \sin (\frac{5 π}{12})$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Apply the sum of angles identity.

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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

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

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

Multiply $2$ by $4$ .

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

Apply the distributive property.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2} \sqrt{2} + \sqrt{2} \sqrt{6}}{8}$

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2 \cdot 2} + \sqrt{2} \sqrt{6}}{8}$

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2 \cdot 2} + \sqrt{2 \cdot 6}}{8}$

Simplify each term.

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{4} + \sqrt{2 \cdot 6}}{8}$

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

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2^{2}} + \sqrt{2 \cdot 6}}{8}$

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

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{2 \cdot 6}}{8}$

Multiply $2$ by $6$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{12}}{8}$

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

Factor $4$ out of $12$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{4 (3)}}{8}$

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

$\frac{\sqrt{3} - 1}{4} + \frac{2 + \sqrt{2^{2} \cdot 3}}{8}$

Pull terms out from under the radical.

$\frac{\sqrt{3} - 1}{4} + \frac{2 + 2 \sqrt{3}}{8}$

Cancel the common factor of $2 + 2 \sqrt{3}$ and $8$ .

Factor $2$ out of $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 \cdot 1 + 2 \sqrt{3}}{8}$

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

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

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

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{3} - 1}{4} + \frac{1 + \sqrt{3}}{4}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - 1 + 1 + \sqrt{3}}{4}$

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

$\frac{- 1 + 1 + 2 \sqrt{3}}{4}$

Simplify by adding numbers.

Add $- 1$ and $1$ .

$\frac{0 + 2 \sqrt{3}}{4}$

Add $0$ and $2 \sqrt{3}$ .

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

Cancel the common factor of $2$ and $4$ .

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

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$

Find the Exact Value sin(pi/4)cos((5pi)/12)+cos(pi/4)sin((5pi)/12)

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