Precalculus Examples

$\sin (\frac{23 π}{4}) \cos (\frac{5 π}{6}) + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 1

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

$\sin (\frac{7 π}{4}) \cos (\frac{5 π}{6}) + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \sin (\frac{π}{4}) \cos (\frac{5 π}{6}) + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 3

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

$- \frac{\sqrt{2}}{2} \cos (\frac{5 π}{6}) + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 4

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.

$- \frac{\sqrt{2}}{2} (- \cos (\frac{π}{6})) + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 5

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

$- \frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2}) + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 6

Multiply $- \frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2})$ .

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Step 6.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 6.2

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

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 6.3

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

$\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 6.4

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 6.5

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 6.6

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} + \cos (\frac{23 π}{4}) \sin (\frac{5 π}{6})$

Step 7

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

$\frac{\sqrt{6}}{4} + \cos (\frac{7 π}{4}) \sin (\frac{5 π}{6})$

Step 8

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

$\frac{\sqrt{6}}{4} + \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 9

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

$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \sin (\frac{5 π}{6})$

Step 10

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

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

Step 11

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 12

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

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Step 12.1

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

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

Step 12.2

Multiply $2$ by $2$ .

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

Step 13

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$