Trigonometry Examples

$\sin (\frac{π}{4} + \frac{π}{3})$

Step 1

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

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

Step 2

Remove parentheses.

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

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

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 3.2

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 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

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 3.5

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 3.6

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

Tap for more steps...

Step 3.6.1

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

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

Step 3.6.2

Combine using the product rule for radicals.

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

Step 3.6.3

Multiply $2$ by $3$ .

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

Step 3.6.4

Multiply $2$ by $2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$