Find the Exact Value sin(165)*sin(15)

$\sin (165) \cdot \sin (15)$

The exact value of $\sin (165)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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

$\sin (15) \cdot \sin (15)$

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\sin (45 - 30) \cdot \sin (15)$

Separate negation.

$\sin (45 - (30)) \cdot \sin (15)$

Apply the difference of angles identity.

$(\sin (45) \cos (30) - \cos (45) \sin (30)) \cdot \sin (15)$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)) \cdot \sin (15)$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)) \cdot \sin (15)$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)) \cdot \sin (15)$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \sin (15)$

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

Simplify each term.

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

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \sin (15)$

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \sin (15)$

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \sin (15)$

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot \sin (15)$

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

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

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cdot \sin (15)$

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cdot \sin (15)$

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \sin (15)$

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \sin (45 - 30)$

Separate negation.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \sin (45 - (30))$

Apply the difference of angles identity.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot (\sin (45) \cos (30) - \cos (45) \sin (30))$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

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

Simplify each term.

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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

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

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

Raise $\sqrt{6} - \sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{6} - \sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Multiply $4$ by $4$ .

$\frac{{(\sqrt{6} - \sqrt{2})}^{2}}{16}$

Rewrite ${(\sqrt{6} - \sqrt{2})}^{2}$ as $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ .

$\frac{(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})}{16}$

Expand $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{6} (\sqrt{6} - \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $6$ by $6$ .

$\frac{\sqrt{36} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

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

$\frac{6 + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $\sqrt{6} (- \sqrt{2})$ .

Combine using the product rule for radicals.

$\frac{6 - \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $6$ by $2$ .

$\frac{6 - \sqrt{12} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

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

Factor $4$ out of $12$ .

$\frac{6 - \sqrt{4 (3)} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

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

$\frac{6 - \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Pull terms out from under the radical.

$\frac{6 - (2 \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $2$ by $- 1$ .

$\frac{6 - 2 \sqrt{3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $- \sqrt{2} \sqrt{6}$ .

Combine using the product rule for radicals.

$\frac{6 - 2 \sqrt{3} - \sqrt{6 \cdot 2} - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $6$ by $2$ .

$\frac{6 - 2 \sqrt{3} - \sqrt{12} - \sqrt{2} (- \sqrt{2})}{16}$

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

Factor $4$ out of $12$ .

$\frac{6 - 2 \sqrt{3} - \sqrt{4 (3)} - \sqrt{2} (- \sqrt{2})}{16}$

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

$\frac{6 - 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - \sqrt{2} (- \sqrt{2})}{16}$

Pull terms out from under the radical.

$\frac{6 - 2 \sqrt{3} - (2 \sqrt{3}) - \sqrt{2} (- \sqrt{2})}{16}$

Multiply $2$ by $- 1$ .

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{2} (- \sqrt{2})}{16}$

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

Multiply $- 1$ by $- 1$ .

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{2} \sqrt{2}}{16}$

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{2} \sqrt{2}}{16}$

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} \sqrt{2}}{16}$

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{16}$

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1 + 1}}{16}$

Add $1$ and $1$ .

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{2}}{16}$

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

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {(2^{\frac{1}{2}})}^{2}}{16}$

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{1}{2} \cdot 2}}{16}$

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

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{16}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{16}$

Rewrite the expression.

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{1}}{16}$

Evaluate the exponent.

$\frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2}{16}$

Add $6$ and $2$ .

$\frac{8 - 2 \sqrt{3} - 2 \sqrt{3}}{16}$

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$\frac{8 - 4 \sqrt{3}}{16}$

Cancel the common factor of $8 - 4 \sqrt{3}$ and $16$ .

Factor $4$ out of $8$ .

$\frac{4 (2) - 4 \sqrt{3}}{16}$

Factor $4$ out of $- 4 \sqrt{3}$ .

$\frac{4 (2) + 4 (- \sqrt{3})}{16}$

Factor $4$ out of $4 (2) + 4 (- \sqrt{3})$ .

$\frac{4 (2 - \sqrt{3})}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.06698729 \dots$

Find the Exact Value sin(165)*sin(15)

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