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Popular >

pi/4-arccos(x)>0

Solution

\frac{π}{4} - arccos (x) > 0

\frac{2}{2} < x \leq 1

Interval Notation

(\frac{2}{2}, 1]

Decimal Notation

0.70710 \dots < x \leq 1

Solution steps

\frac{π}{4} - arccos (x) > 0

Move

\frac{π}{4}

to the right side

\frac{π}{4} - arccos (x) > 0

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} - arccos (x) - \frac{π}{4} > 0 - \frac{π}{4}

Simplify

- arccos (x) > - \frac{π}{4}

- arccos (x) > - \frac{π}{4}

Multiply both sides by

- 1

- arccos (x) > - \frac{π}{4}

Multiply both sides by -1 (reverse the inequality)

(- arccos (x)) (- 1) < (- \frac{π}{4}) (- 1)

Simplify

arccos (x) < \frac{π}{4}

arccos (x) < \frac{π}{4}

arccos (x) < a

then

x > cos (a)

x > cos (\frac{π}{4})

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

x > \frac{2}{2}

Domain of

arccos (x) : - 1 \leq x \leq 1

Domain definition

Find known functions domain restrictions:

- 1 \leq x \leq 1

arccos (f (x)) \Rightarrow - 1 \leq f (x) \leq 1

Solve

- 1 \leq x \leq 1 : - 1 \leq x \leq 1

- 1 \leq x \leq 1

Could not simplify further

- 1 \leq x \leq 1

The function domain

- 1 \leq x \leq 1

Combine the intervals

x > \frac{2}{2} and - 1 \leq x \leq 1

Merge Overlapping Intervals

x > \frac{2}{2} and - 1 \leq x \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

x > \frac{2}{2}

and

- 1 \leq x \leq 1

\frac{2}{2} < x \leq 1

\frac{2}{2} < x \leq 1

tan (θ) > 0, csc (θ) > 0

3 cos (t) \geq 0

sin (x) > \frac{1}{3}

cos (x) - \frac{2}{2} \leq 0

4 \cdot sin^{2} (X) - 2 < 0