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Popular >

pi/2-arctan(n)<= 10^{-3}

Solution

\frac{π}{2} - arctan (n) \leq 1 0^{- 3}

Solution

n \geq tan (- \frac{1}{1000} + \frac{π}{2})

+2

Interval Notation

[tan (- \frac{1}{1000} + \frac{π}{2}), \infty)

Decimal Notation

n \geq 999.99966 \dots

Solution steps

\frac{π}{2} - arctan (n) \leq 1 0^{- 3}

Simplify

1 0^{- 3} : \frac{1}{1000}

1 0^{- 3}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{1 0 ^{3}}

1 0^{3} = 1000

= \frac{1}{1000}

\frac{π}{2} - arctan (n) \leq \frac{1}{1000}

Move

\frac{π}{2}

to the right side

\frac{π}{2} - arctan (n) \leq \frac{1}{1000}

Subtract

\frac{π}{2}

from both sides

\frac{π}{2} - arctan (n) - \frac{π}{2} \leq \frac{1}{1000} - \frac{π}{2}

Simplify

- arctan (n) \leq \frac{1}{1000} - \frac{π}{2}

- arctan (n) \leq \frac{1}{1000} - \frac{π}{2}

Multiply both sides by

- 1

- arctan (n) \leq \frac{1}{1000} - \frac{π}{2}

Multiply both sides by -1 (reverse the inequality)

(- arctan (n)) (- 1) \geq \frac{1}{1000} (- 1) - \frac{π}{2} (- 1)

Simplify

arctan (n) \geq - \frac{1}{1000} + \frac{π}{2}

arctan (n) \geq - \frac{1}{1000} + \frac{π}{2}

If

arctan (x) \geq a

then

x \geq tan (a)

n \geq tan (- \frac{1}{1000} + \frac{π}{2})

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