Finite Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $2$ .

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

Step 1.5

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

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Step 1.5.1

Factor $2$ out of $2 π$ .

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.6

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

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

Step 1.7

Multiply $\sin (\frac{π}{4})$ by $1$ .

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

Step 1.8

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

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

Step 2

To write $\frac{\sqrt{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.35355339 \dots$