Trigonometry Examples

$\sin (\frac{5 π}{4} - \frac{π}{6})$

Step 1

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

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

Step 2

Remove parentheses.

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

Step 3

Simplify each term.

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Step 3.1

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 third quadrant.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Combine using the product rule for radicals.

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

Step 3.4.3

Multiply $3$ by $2$ .

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

Step 3.4.4

Multiply $2$ by $2$ .

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

Step 3.5

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 third quadrant.

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

Step 3.6

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 3.7

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

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{\sqrt{6}}{4} + 1 \frac{\sqrt{2}}{2} \sin (\frac{π}{6})$

Step 3.7.2

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

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

Step 3.8

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 3.9

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

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Step 3.9.1

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

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

Step 3.9.2

Multiply $2$ by $2$ .

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

Step 4

Simplify.

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Step 4.1

Combine the numerators over the common denominator.

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

Step 4.2

Factor $- 1$ out of $- \sqrt{6}$ .

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

Step 4.3

Factor $- 1$ out of $\sqrt{2}$ .

$\frac{- (\sqrt{6}) - 1 (- \sqrt{2})}{4}$

Step 4.4

Factor $- 1$ out of $- (\sqrt{6}) - 1 (- \sqrt{2})$ .

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

Step 4.5

Simplify the expression.

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Step 4.5.1

Rewrite $- (\sqrt{6} - \sqrt{2})$ as $- 1 (\sqrt{6} - \sqrt{2})$ .

$\frac{- 1 (\sqrt{6} - \sqrt{2})}{4}$

Step 4.5.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$