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Popular >

1<= sin(θ)<3

Solution

1 \leq sin (θ) < 3

Solution

θ = \frac{π}{2} + 2 πn

Decimal Notation

θ = 1.57079 \dots + 2 πn

Solution steps

1 \leq sin (θ) < 3

a \leq u < b

then

a \leq u and u < b

1 \leq sin (θ) and sin (θ) < 3

1 \leq sin (θ) : θ = \frac{π}{2} + 2 πn

1 \leq sin (θ)

Switch sides

sin (θ) \geq 1

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (1) + 2 πn \leq θ \leq π - arcsin (1) + 2 πn

Simplify

arcsin (1) : \frac{π}{2}

arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

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}

= \frac{π}{2}

Simplify

π - arcsin (1) : \frac{π}{2}

π - arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

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 - \frac{π}{2}

Simplify

π - \frac{π}{2}

Convert element to fraction:

π = \frac{π 2}{2}

= \frac{π 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{2}

Add similar elements:

2 π - π = π

= \frac{π}{2}

= \frac{π}{2}

\frac{π}{2} + 2 πn \leq θ \leq \frac{π}{2} + 2 πn

Simplify

θ = \frac{π}{2} + 2 πn

sin (θ) < 3 :

True for all

θ \in R

sin (θ) < 3

Range of

sin (θ) : - 1 \leq sin (θ) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (θ) \leq 1

- 1 \leq sin (θ) \leq 1

sin (θ) < 3 and - 1 \leq sin (θ) \leq 1 : - 1 \leq sin (θ) \leq 1

Let

y = sin (θ)

Combine the intervals

y < 3 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < 3 and - 1 \leq y \leq 1

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

y < 3

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all θ

True for all θ \in R

Combine the intervals

θ = \frac{π}{2} + 2 πn and True for all θ \in R

Merge Overlapping Intervals

θ = \frac{π}{2} + 2 πn

Popular Examples

cos(θ)>0\land sin(θ)<0

cos (θ) > 0 and sin (θ) < 0

tan(θ)=1\land cos(θ)>0,csc(θ)

tan (θ) = 1 and cos (θ) > 0, csc (θ)

-1<sin(x)<= 0

- 1 < sin (x) \leq 0

tan(θ)= 1/3 \land sin(θ)<0

tan (θ) = \frac{1}{3} and sin (θ) < 0

cot(θ)=2\land sec(θ)>= 0

cot (θ) = 2 and sec (θ) \geq 0