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Popular Trigonometry >

pi/2-arccos(x-1)>0

Solution

\frac{π}{2} - arccos (x - 1) > 0

Solution

1 < x \leq 2

Interval Notation

(1, 2]

Solution steps

\frac{π}{2} - arccos (x - 1) > 0

Move

\frac{π}{2}

to the right side

\frac{π}{2} - arccos (x - 1) > 0

Subtract

\frac{π}{2}

from both sides

\frac{π}{2} - arccos (x - 1) - \frac{π}{2} > 0 - \frac{π}{2}

Simplify

- arccos (x - 1) > - \frac{π}{2}

- arccos (x - 1) > - \frac{π}{2}

Multiply both sides by

- 1

- arccos (x - 1) > - \frac{π}{2}

Multiply both sides by -1 (reverse the inequality)

(- arccos (x - 1)) (- 1) < (- \frac{π}{2}) (- 1)

Simplify

arccos (x - 1) < \frac{π}{2}

arccos (x - 1) < \frac{π}{2}

arccos (x) < a

then

x > cos (a)

x - 1 > cos (\frac{π}{2})

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

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}

= 0

x - 1 > 0

x - 1 > 0 : x > 1

x - 1 > 0

Move

1

to the right side

x - 1 > 0

Add

1

to both sides

x - 1 + 1 > 0 + 1

Simplify

x > 1

x > 1

x > 1

Domain of

arccos (x - 1) : 0 \leq x \leq 2

Domain definition

Find known functions domain restrictions:

0 \leq x \leq 2

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

Solve

- 1 \leq (x - 1) \leq 1 : 0 \leq x \leq 2

- 1 \leq (x - 1) \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

- 1 \leq (x - 1) and (x - 1) \leq 1

- 1 \leq x - 1 : x \geq 0

- 1 \leq x - 1

Add

1

to both sides

- 1 + 1 \leq x - 1 + 1

Simplify

0 \leq x

Switch sides

x \geq 0

x - 1 \leq 1 : x \leq 2

x - 1 \leq 1

Move

1

to the right side

x - 1 \leq 1

Add

1

to both sides

x - 1 + 1 \leq 1 + 1

Simplify

x \leq 2

x \leq 2

Combine the intervals

x \geq 0 and x \leq 2

Merge Overlapping Intervals

x \geq 0 and x \leq 2

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

x \geq 0

and

x \leq 2

0 \leq x \leq 2

0 \leq x \leq 2

The function domain

0 \leq x \leq 2

Combine the intervals

x > 1 and 0 \leq x \leq 2

Merge Overlapping Intervals

x > 1 and 0 \leq x \leq 2

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

x > 1

and

0 \leq x \leq 2

1 < x \leq 2

1 < x \leq 2

Graph

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