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0<= sin(x)<= 1

Solution

0 \leq sin (x) \leq 1

2 πn \leq x \leq π + 2 πn

Interval Notation

[2 πn, π + 2 πn]

Decimal Notation

2 πn \leq x \leq 3.14159 \dots + 2 πn

Solution steps

0 \leq sin (x) \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

0 \leq sin (x) and sin (x) \leq 1

0 \leq sin (x) : 2 πn \leq x \leq π + 2 πn

0 \leq sin (x)

Switch sides

sin (x) \geq 0

For

sin (x) \geq a

, if

- 1 < a < 1

then

arcsin (a) + 2 πn \leq x \leq π - arcsin (a) + 2 πn

arcsin (0) + 2 πn \leq x \leq π - arcsin (0) + 2 πn

Simplify

arcsin (0) : 0

arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0

Simplify

π - arcsin (0) : π

π - arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - 0

π - 0 = π

= π

0 + 2 πn \leq x \leq π + 2 πn

Simplify

2 πn \leq x \leq π + 2 πn

sin (x) \leq 1 :

True for all

x \in R

sin (x) \leq 1

Range of

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

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

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

Let

y = sin (x)

Combine the intervals

y \leq 1 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq 1 and - 1 \leq y \leq 1

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

y \leq 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

2 πn \leq x \leq π + 2 πn and True for all x \in R

Merge Overlapping Intervals

2 πn \leq x \leq π + 2 πn

- \frac{π}{2} < arcsin (y) < \frac{π}{2}

sin (θ) = - \frac{1}{2} and cos (θ) > 0

- \frac{1}{2} < sin (x) < \frac{1}{2}

sin (θ) > 0 and sec (θ) < 0

0 \leq arctan (x) \leq 1