Evaluate 10sin(pi/8)*cos((7pi)/8)

$10 \sin (\frac{π}{8}) \cdot \cos (\frac{7 π}{8})$

The exact value of $\sin (\frac{π}{8})$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$10 \sin (\frac{\frac{π}{4}}{2}) \cdot \cos (\frac{7 π}{8})$

Apply the sine half-angle identity.

$10 (\pm \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}) \cdot \cos (\frac{7 π}{8})$

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

$10 \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}} \cdot \cos (\frac{7 π}{8})$

Simplify $\sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}$ .

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

$10 \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \cdot \cos (\frac{7 π}{8})$

Write $1$ as a fraction with a common denominator.

$10 \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} \cdot \cos (\frac{7 π}{8})$

Combine the numerators over the common denominator.

$10 \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} \cdot \cos (\frac{7 π}{8})$

Multiply the numerator by the reciprocal of the denominator.

$10 \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} \cdot \cos (\frac{7 π}{8})$

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

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

$10 \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} \cdot \cos (\frac{7 π}{8})$

Multiply $2$ by $2$ .

$10 \sqrt{\frac{2 - \sqrt{2}}{4}} \cdot \cos (\frac{7 π}{8})$

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$10 \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} \cdot \cos (\frac{7 π}{8})$

Simplify the denominator.

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

$10 \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} \cdot \cos (\frac{7 π}{8})$

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

$10 \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \cos (\frac{7 π}{8})$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$2 (5) \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \cos (\frac{7 π}{8})$

Cancel the common factor.

$2 \cdot 5 \frac{\sqrt{2 - \sqrt{2}}}{2} \cdot \cos (\frac{7 π}{8})$

Rewrite the expression.

$5 \sqrt{2 - \sqrt{2}} \cdot \cos (\frac{7 π}{8})$

The exact value of $\cos (\frac{7 π}{8})$ is $- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Rewrite $\frac{7 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$5 \sqrt{2 - \sqrt{2}} \cdot \cos (\frac{\frac{7 π}{4}}{2})$

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$5 \sqrt{2 - \sqrt{2}} \cdot (\pm \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}})$

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}})$

Simplify $- \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}}$ .

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

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

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

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})$

Write $1$ as a fraction with a common denominator.

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}})$

Combine the numerators over the common denominator.

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}})$

Multiply the numerator by the reciprocal of the denominator.

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}})$

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

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

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}})$

Multiply $2$ by $2$ .

$5 \sqrt{2 - \sqrt{2}} \cdot (- \sqrt{\frac{2 + \sqrt{2}}{4}})$

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$5 \sqrt{2 - \sqrt{2}} \cdot (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})$

Simplify the denominator.

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

$5 \sqrt{2 - \sqrt{2}} \cdot (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}})$

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

$5 \sqrt{2 - \sqrt{2}} \cdot (- \frac{\sqrt{2 + \sqrt{2}}}{2})$

Multiply $5 \sqrt{2 - \sqrt{2}} (- \frac{\sqrt{2 + \sqrt{2}}}{2})$ .

Multiply $- 1$ by $5$ .

$- 5 \sqrt{2 - \sqrt{2}} \frac{\sqrt{2 + \sqrt{2}}}{2}$

Combine $\frac{\sqrt{2 + \sqrt{2}}}{2}$ and $- 5$ .

$\frac{\sqrt{2 + \sqrt{2}} \cdot - 5}{2} \sqrt{2 - \sqrt{2}}$

Combine $\frac{\sqrt{2 + \sqrt{2}} \cdot - 5}{2}$ and $\sqrt{2 - \sqrt{2}}$ .

$\frac{\sqrt{2 + \sqrt{2}} \cdot - 5 \sqrt{2 - \sqrt{2}}}{2}$

Combine using the product rule for radicals.

$\frac{- 5 \sqrt{(2 + \sqrt{2}) (2 - \sqrt{2})}}{2}$

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

Apply the distributive property.

$\frac{- 5 \sqrt{2 (2 - \sqrt{2}) + \sqrt{2} (2 - \sqrt{2})}}{2}$

Apply the distributive property.

$\frac{- 5 \sqrt{2 \cdot 2 + 2 (- \sqrt{2}) + \sqrt{2} (2 - \sqrt{2})}}{2}$

Apply the distributive property.

$\frac{- 5 \sqrt{2 \cdot 2 + 2 (- \sqrt{2}) + \sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$\frac{- 5 \sqrt{4 + 2 (- \sqrt{2}) + \sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}}{2}$

Multiply $- 1$ by $2$ .

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} (- \sqrt{2})}}{2}$

Move $2$ to the left of $\sqrt{2}$ .

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \cdot \sqrt{2} + \sqrt{2} (- \sqrt{2})}}{2}$

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

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

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}}{2}$

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

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}}{2}$

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

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - {\sqrt{2}}^{1 + 1}}}{2}$

Add $1$ and $1$ .

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - {\sqrt{2}}^{2}}}{2}$

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

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

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}}{2}$

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

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}}{2}$

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

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{\frac{2}{2}}}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{\frac{2}{2}}}}{2}$

Rewrite the expression.

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2^{1}}}{2}$

Evaluate the exponent.

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 1 \cdot 2}}{2}$

Multiply $- 1$ by $2$ .

$\frac{- 5 \sqrt{4 - 2 \sqrt{2} + 2 \sqrt{2} - 2}}{2}$

Subtract $2$ from $4$ .

$\frac{- 5 \sqrt{2 - 2 \sqrt{2} + 2 \sqrt{2}}}{2}$

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

$\frac{- 5 \sqrt{2 + 0}}{2}$

Add $2$ and $0$ .

$\frac{- 5 \sqrt{2}}{2}$

Move the negative in front of the fraction.

$- \frac{5 \sqrt{2}}{2}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 \sqrt{2}}{2}$

Decimal Form:

$- 3.53553390 \dots$

Evaluate 10sin(pi/8)*cos((7pi)/8)

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