Trigonometry Examples

$\sin (\frac{19 π}{4}) \cos (\frac{π}{6}) + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (\frac{3 π}{4}) \cos (\frac{π}{6}) + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 2

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

$\sin (\frac{π}{4}) \cos (\frac{π}{6}) + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 3

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

$\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 4

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 5

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

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Step 5.1

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

$\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 5.2

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 5.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 5.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} + \cos (\frac{19 π}{4}) \sin (\frac{π}{6})$

Step 6

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 7

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.

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

Step 8

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

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

Step 9

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

Multiply $2$ by $2$ .

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$