Simplify sin(45 degrees )cos(30 degrees )+cos(45 degrees )sin(30 degrees )

$\sin (45 °) \cos (30 °) + \cos (45 °) \sin (30 °)$

Step 1

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

$\frac{\sqrt{2}}{2} \cos (30 °) + \cos (45 °) \sin (30 °)$

Step 2

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

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \cos (45 °) \sin (30 °)$

Step 3

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

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

$\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \cos (45 °) \sin (30 °)$

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \cos (45 °) \sin (30 °)$

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \cos (45 °) \sin (30 °)$

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} + \cos (45 °) \sin (30 °)$

Step 4

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

$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \sin (30 °)$

Step 5

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

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

Step 6

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

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

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

Multiply $2$ by $2$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$