Find the Exact Value sin(pi/4)cos(pi/3)+cos(pi/4)sin(pi/3)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

Multiply $2$ by $2$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $2$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$