Finite Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Multiply $3$ by $2$ .

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

Step 4.4

Multiply $2$ by $2$ .

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

Step 5

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.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $2$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$