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pi/(14)-1/7 arctan(x/7)<= 0.0002

Solution

\frac{π}{14} - \frac{1}{7} arctan (\frac{x}{7}) \leq 0.0002

x \geq 4999.99673 \dots

Interval Notation

[4999.99673 \dots, \infty)

Solution steps

\frac{π}{14} - \frac{1}{7} arctan (\frac{x}{7}) \leq 0.0002

Move

\frac{π}{14}

to the right side

\frac{π}{14} - \frac{1}{7} arctan (\frac{x}{7}) \leq 0.0002

Subtract

\frac{π}{14}

from both sides

\frac{π}{14} - \frac{1}{7} arctan (\frac{x}{7}) - \frac{π}{14} \leq 0.0002 - \frac{π}{14}

Simplify

- \frac{1}{7} arctan (\frac{x}{7}) \leq 0.0002 - \frac{π}{14}

- \frac{1}{7} arctan (\frac{x}{7}) \leq 0.0002 - \frac{π}{14}

Multiply both sides by

- 1

- \frac{1}{7} arctan (\frac{x}{7}) \leq 0.0002 - \frac{π}{14}

Multiply both sides by -1 (reverse the inequality)

(- \frac{1}{7} arctan (\frac{x}{7})) (- 1) \geq 0.0002 (- 1) - \frac{π}{14} (- 1)

Simplify

\frac{1}{7} arctan (\frac{x}{7}) \geq - 0.0002 + \frac{π}{14}

\frac{1}{7} arctan (\frac{x}{7}) \geq - 0.0002 + \frac{π}{14}

Multiply both sides by

7

\frac{1}{7} arctan (\frac{x}{7}) \geq - 0.0002 + \frac{π}{14}

Multiply both sides by

7

7 \cdot \frac{1}{7} arctan (\frac{x}{7}) \geq - 7 \cdot 0.0002 + 7 \cdot \frac{π}{14}

Simplify

7 \cdot \frac{1}{7} arctan (\frac{x}{7}) \geq - 7 \cdot 0.0002 + 7 \cdot \frac{π}{14}

Simplify

7 \cdot \frac{1}{7} arctan (\frac{x}{7}) : arctan (\frac{x}{7})

7 \cdot \frac{1}{7} arctan (\frac{x}{7})

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 7}{7} arctan (\frac{x}{7})

Cancel the common factor:

7

= arctan (\frac{x}{7}) \cdot 1

Multiply:

arctan (\frac{x}{7}) \cdot 1 = arctan (\frac{x}{7})

= arctan (\frac{x}{7})

Simplify

- 7 \cdot 0.0002 + 7 \cdot \frac{π}{14} : - 0.0014 + \frac{π}{2}

- 7 \cdot 0.0002 + 7 \cdot \frac{π}{14}

7 \cdot 0.0002 = 0.0014

7 \cdot 0.0002

Multiply the numbers:

7 \cdot 0.0002 = 0.0014

= 0.0014

7 \cdot \frac{π}{14} = \frac{π}{2}

7 \cdot \frac{π}{14}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 7}{14}

Cancel the common factor:

7

= \frac{π}{2}

= - 0.0014 + \frac{π}{2}

arctan (\frac{x}{7}) \geq - 0.0014 + \frac{π}{2}

arctan (\frac{x}{7}) \geq - 0.0014 + \frac{π}{2}

arctan (\frac{x}{7}) \geq - 0.0014 + \frac{π}{2}

arctan (x) \geq a

then

x \geq tan (a)

\frac{x}{7} \geq tan (- 0.0014 + \frac{π}{2})

tan (- 0.0014 + \frac{π}{2}) = 714.28524 \dots

tan (- 0.0014 + \frac{π}{2})

tan (- 0.0014 + \frac{π}{2}) = 714.28524 \dots

= 714.28524 \dots

\frac{x}{7} \geq 714.28524 \dots

\frac{x}{7} \geq 714.28524 \dots : x \geq 4999.99673 \dots

\frac{x}{7} \geq 714.28524 \dots

Multiply both sides by

7

\frac{x}{7} \geq 714.28524 \dots

Multiply both sides by

7

\frac{7 x}{7} \geq 714.28524 \dots \cdot 7

Simplify

x \geq 4999.99673 \dots

x \geq 4999.99673 \dots

x \geq 4999.99673 \dots

- sin (2 x - \frac{π}{12}) \leq \frac{2}{2}

\frac{1}{cos ( x )} \leq 0

sin^{2} (\frac{x}{2}) + cos (x) > 0

sin^{2} (x) \geq \frac{1}{2}

cos (x) > - \frac{3}{2}