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Popular >

pi/2-arctan(n^2)<= 0.001

Solution

\frac{π}{2} - arctan (n^{2}) \leq 0.001

Solution

n \leq - 31.62277 \dots or n \geq 31.62277 \dots

Interval Notation

(- \infty, - 31.62277 \dots] \cup [31.62277 \dots, \infty)

Solution steps

\frac{π}{2} - arctan (n^{2}) \leq 0.001

Move

\frac{π}{2}

to the right side

\frac{π}{2} - arctan (n^{2}) \leq 0.001

Subtract

\frac{π}{2}

from both sides

\frac{π}{2} - arctan (n^{2}) - \frac{π}{2} \leq 0.001 - \frac{π}{2}

Simplify

- arctan (n^{2}) \leq 0.001 - \frac{π}{2}

- arctan (n^{2}) \leq 0.001 - \frac{π}{2}

Multiply both sides by

- 1

- arctan (n^{2}) \leq 0.001 - \frac{π}{2}

Multiply both sides by -1 (reverse the inequality)

(- arctan (n^{2})) (- 1) \geq 0.001 (- 1) - \frac{π}{2} (- 1)

Simplify

arctan (n^{2}) \geq - 0.001 + \frac{π}{2}

arctan (n^{2}) \geq - 0.001 + \frac{π}{2}

arctan (x) \geq a

then

x \geq tan (a)

n^{2} \geq tan (- 0.001 + \frac{π}{2})

tan (- 0.001 + \frac{π}{2}) = 999.99966 \dots

tan (- 0.001 + \frac{π}{2})

tan (- 0.001 + \frac{π}{2}) = 999.99966 \dots

= 999.99966 \dots

n^{2} \geq 999.99966 \dots

n^{2} \geq 999.99966 \dots : n \leq - 31.62277 \dots or n \geq 31.62277 \dots

n^{2} \geq 999.99966 \dots

Rewrite in standard form

n^{2} \geq 999.99966 \dots

Subtract

999.99966 \dots

from both sides

n^{2} - 999.99966 \dots \geq 999.99966 \dots - 999.99966 \dots

Simplify

n^{2} - 999.99966 \dots \geq 0

n^{2} - 999.99966 \dots \geq 0

Factor

n^{2} - 999.99966 \dots : (n + 999.99966 \dots) (n - 999.99966 \dots)

n^{2} - 999.99966 \dots

Apply radical rule:

a = (a)^{2}

999.99966 \dots = (999.99966 \dots)^{2}

= n^{2} - (999.99966 \dots)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

n^{2} - (999.99966 \dots)^{2} = (n + 999.99966 \dots) (n - 999.99966 \dots)

= (n + 999.99966 \dots) (n - 999.99966 \dots)

(n + 999.99966 \dots) (n - 999.99966 \dots) \geq 0

Identify the intervals

Find the signs of the factors of

(n + 999.99966 \dots) (n - 999.99966 \dots)

Find the signs of

n + 999.99966 \dots

n + 999.99966 \dots = 0 : n = - 31.62277 \dots

n + 999.99966 \dots = 0

Move

999.99966 \dots

to the right side

n + 999.99966 \dots = 0

Subtract

999.99966 \dots

from both sides

n + 999.99966 \dots - 999.99966 \dots = 0 - 999.99966 \dots

Simplify

n = - 999.99966 \dots

n = - 999.99966 \dots

Simplify

n = - 31.62277 \dots

n + 999.99966 \dots < 0 : n < - 31.62277 \dots

n + 999.99966 \dots < 0

Move

999.99966 \dots

to the right side

n + 999.99966 \dots < 0

Subtract

999.99966 \dots

from both sides

n + 999.99966 \dots - 999.99966 \dots < 0 - 999.99966 \dots

Simplify

n < - 999.99966 \dots

n < - 999.99966 \dots

Simplify

n < - 31.62277 \dots

n + 999.99966 \dots > 0 : n > - 31.62277 \dots

n + 999.99966 \dots > 0

Move

999.99966 \dots

to the right side

n + 999.99966 \dots > 0

Subtract

999.99966 \dots

from both sides

n + 999.99966 \dots - 999.99966 \dots > 0 - 999.99966 \dots

Simplify

n > - 999.99966 \dots

n > - 999.99966 \dots

Simplify

n > - 31.62277 \dots

Find the signs of

n - 999.99966 \dots

n - 999.99966 \dots = 0 : n = 31.62277 \dots

n - 999.99966 \dots = 0

Move

999.99966 \dots

to the right side

n - 999.99966 \dots = 0

Add

999.99966 \dots

to both sides

n - 999.99966 \dots + 999.99966 \dots = 0 + 999.99966 \dots

Simplify

n = 999.99966 \dots

n = 999.99966 \dots

Simplify

n = 31.62277 \dots

n - 999.99966 \dots < 0 : n < 31.62277 \dots

n - 999.99966 \dots < 0

Move

999.99966 \dots

to the right side

n - 999.99966 \dots < 0

Add

999.99966 \dots

to both sides

n - 999.99966 \dots + 999.99966 \dots < 0 + 999.99966 \dots

Simplify

n < 999.99966 \dots

n < 999.99966 \dots

Simplify

n < 31.62277 \dots

n - 999.99966 \dots > 0 : n > 31.62277 \dots

n - 999.99966 \dots > 0

Move

999.99966 \dots

to the right side

n - 999.99966 \dots > 0

Add

999.99966 \dots

to both sides

n - 999.99966 \dots + 999.99966 \dots > 0 + 999.99966 \dots

Simplify

n > 999.99966 \dots

n > 999.99966 \dots

Simplify

n > 31.62277 \dots

Summarize in a table:

n + 999.99966 \dots n - 999.99966 \dots (n + 999.99966 \dots) (n - 999.99966 \dots) n < - 31.62277 \dots - - + n = - 31.62277 \dots 0 - 0 - 31.62277 \dots < n < 31.62277 \dots + - - n = 31.62277 \dots + 00 n > 31.62277 \dots + + +

Identify the intervals that satisfy the required condition:

\geq 0

n < - 31.62277 \dots or n = - 31.62277 \dots or n = 31.62277 \dots or n > 31.62277 \dots

Merge Overlapping Intervals

n \leq - 31.62277 \dots or n = 31.62277 \dots or n > 31.62277 \dots

The union of two intervals is the set of numbers which are in either interval

n < - 31.62277 \dots

n = - 31.62277 \dots

n \leq - 31.62277 \dots

The union of two intervals is the set of numbers which are in either interval

n \leq - 31.62277 \dots

n = 31.62277 \dots

n \leq - 31.62277 \dots or n = 31.62277 \dots

The union of two intervals is the set of numbers which are in either interval

n \leq - 31.62277 \dots or n = 31.62277 \dots

n > 31.62277 \dots

n \leq - 31.62277 \dots or n \geq 31.62277 \dots

n \leq - 31.62277 \dots or n \geq 31.62277 \dots

n \leq - 31.62277 \dots or n \geq 31.62277 \dots

n \leq - 31.62277 \dots or n \geq 31.62277 \dots

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