Precalculus Examples

$\sin (- \frac{π}{4} - 1000 π)$

Step 1

To write $- 1000 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sin (- \frac{π}{4} - 1000 π \cdot \frac{4}{4})$

Step 2

Combine fractions.

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Step 2.1

Combine $- 1000 π$ and $\frac{4}{4}$ .

$\sin (- \frac{π}{4} + \frac{- 1000 π \cdot 4}{4})$

Step 2.2

Combine the numerators over the common denominator.

$\sin (\frac{- π - 1000 π \cdot 4}{4})$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $4$ by $- 1000$ .

$\sin (\frac{- π - 4000 π}{4})$

Step 3.2

Subtract $4000 π$ from $- π$ .

$\sin (\frac{- 4001 π}{4})$

Step 4

Move the negative in front of the fraction.

$\sin (- \frac{4001 π}{4})$

Step 5

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

$\sin (\frac{7 π}{4})$

Step 6

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.

$- \sin (\frac{π}{4})$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.70710678 \dots$