English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(x)+cos(x)>= 0

Solution

sin (x) + cos (x) \geq 0

Solution

- \frac{π}{4} + 2 πn \leq x \leq \frac{3 π}{4} + 2 πn

Interval Notation

[- \frac{π}{4} + 2 πn, \frac{3 π}{4} + 2 πn]

Decimal Notation

- 0.78539 \dots + 2 πn \leq x \leq 2.35619 \dots + 2 πn

Solution steps

sin (x) + cos (x) \geq 0

Use the following identity:

cos (x) + sin (x) = 2 sin (\frac{π}{4} + x)

2 sin (\frac{π}{4} + x) \geq 0

Divide both sides by

2

2 sin (\frac{π}{4} + x) \geq 0

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} + x )}{2} \geq \frac{0}{2}

Simplify

sin (\frac{π}{4} + x) \geq 0

sin (\frac{π}{4} + 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 (\frac{π}{4} + x) \leq π - arcsin (0) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (0) + 2 πn \leq \frac{π}{4} + x and \frac{π}{4} + x \leq π - arcsin (0) + 2 πn

arcsin (0) + 2 πn \leq \frac{π}{4} + x : x \geq 2 πn - \frac{π}{4}

arcsin (0) + 2 πn \leq \frac{π}{4} + x

Switch sides

\frac{π}{4} + x \geq arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

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 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

\frac{π}{4} + x \geq 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x \geq 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} + x - \frac{π}{4} \geq 2 πn - \frac{π}{4}

Simplify

x \geq 2 πn - \frac{π}{4}

x \geq 2 πn - \frac{π}{4}

\frac{π}{4} + x \leq π - arcsin (0) + 2 πn : x \leq \frac{3 π}{4} + 2 πn

\frac{π}{4} + x \leq π - arcsin (0) + 2 πn

Simplify

π - arcsin (0) + 2 πn : π + 2 πn

π - arcsin (0) + 2 πn

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 + 2 πn

π - 0 + 2 πn = π + 2 πn

= π + 2 πn

\frac{π}{4} + x \leq π + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x \leq π + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} + x - \frac{π}{4} \leq π + 2 πn - \frac{π}{4}

Simplify

x \leq π + 2 πn - \frac{π}{4}

x \leq π + 2 πn - \frac{π}{4}

Simplify

π - \frac{π}{4} : \frac{3 π}{4}

π - \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= \frac{π 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 - π}{4}

Add similar elements:

4 π - π = 3 π

= \frac{3 π}{4}

x \leq \frac{3 π}{4} + 2 πn

Combine the intervals

x \geq 2 πn - \frac{π}{4} and x \leq \frac{3 π}{4} + 2 πn

Merge Overlapping Intervals

- \frac{π}{4} + 2 πn \leq x \leq \frac{3 π}{4} + 2 πn

Popular Examples

tan(θ)<0,sin(θ)<0

tan (θ) < 0, sin (θ) < 0

sin(θ)>0,cos(θ)>0

sin (θ) > 0, cos (θ) > 0

sec(θ)<0

sec (θ) < 0

cos(x)>= (sqrt(2))/2

cos (x) \geq \frac{2}{2}

cos(2x)<0

cos (2 x) < 0