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Popular >

cos(x)<sin(2x)

Solution

cos (x) < sin (2 x)

Solution

\frac{π}{6} + 2 πn < x < \frac{π}{2} + 2 πn or \frac{5 π}{6} + 2 πn < x < \frac{3 π}{2} + 2 πn

Interval Notation

(\frac{π}{6} + 2 πn, \frac{π}{2} + 2 πn) \cup (\frac{5 π}{6} + 2 πn, \frac{3 π}{2} + 2 πn)

Decimal Notation

0.52359 \dots + 2 πn < x < 1.57079 \dots + 2 πn or 2.61799 \dots + 2 πn < x < 4.71238 \dots + 2 πn

Solution steps

cos (x) < sin (2 x)

Move

sin (2 x)

to the left side

cos (x) < sin (2 x)

Subtract

sin (2 x)

from both sides

cos (x) - sin (2 x) < sin (2 x) - sin (2 x)

cos (x) - sin (2 x) < 0

cos (x) - sin (2 x) < 0

Use the following identity:

sin (2 x) = 2 cos (x) sin (x)

cos (x) - 2 cos (x) sin (x) < 0

Periodicity of

cos (x) - 2 cos (x) sin (x) : 2 π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

cos (x), 2 cos (x) sin (x)

Periodicity of

cos (x) : 2 π

Periodicity of

cos (x)

2 π

= 2 π

Periodicity of

2 cos (x) sin (x) : π

2 cos (x) sin (x)

is composed of the following functions and periods:

cos (x)

with periodicity of

2 π

The compound periodicity is:

π

Combine periods:

2 π, π

= 2 π

Factor

cos (x) - 2 cos (x) sin (x) : - cos (x) (2 sin (x) - 1)

cos (x) - 2 cos (x) sin (x)

Factor out common term

- cos (x)

= - cos (x) (- 1 + 2 sin (x))

- cos (x) (2 sin (x) - 1) < 0

To find the zeroes, set the inequality to zero

- cos (x) (2 sin (x) - 1) = 0

Solve

- cos (x) (2 sin (x) - 1) = 0

for

0 \leq x < 2 π

- cos (x) (2 sin (x) - 1) = 0

Solving each part separately

cos (x) = 0 : x = \frac{π}{2} or x = \frac{3 π}{2}

cos (x) = 0, 0 \leq x < 2 π

General solutions for

cos (x) = 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}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

2 sin (x) - 1 = 0 : x = \frac{π}{6} or x = \frac{5 π}{6}

2 sin (x) - 1 = 0, 0 \leq x < 2 π

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

2 sin (x) - 1 + 1 = 0 + 1

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{6}, x = \frac{5 π}{6}

Combine all the solutions

\frac{π}{6} or \frac{π}{2} or \frac{5 π}{6} or \frac{3 π}{2}

The intervals between the zeros

0 < x < \frac{π}{6}, \frac{π}{6} < x < \frac{π}{2}, \frac{π}{2} < x < \frac{5 π}{6}, \frac{5 π}{6} < x < \frac{3 π}{2}, \frac{3 π}{2} < x < 2 π

Summarize in a table:

cos (x) 2 sin (x) - 1 - cos (x) (2 sin (x) - 1) x = 0 + - + 0 < x < \frac{π}{6} + - + x = \frac{π}{6} + 00 \frac{π}{6} < x < \frac{π}{2} + + - x = \frac{π}{2} 0 + 0 \frac{π}{2} < x < \frac{5 π}{6} - + + x = \frac{5 π}{6} - 00 \frac{5 π}{6} < x < \frac{3 π}{2} - - - x = \frac{3 π}{2} 0 - 0 \frac{3 π}{2} < x < 2 π + - + x = 2 π + - +

Identify the intervals that satisfy the required condition:

< 0

\frac{π}{6} < x < \frac{π}{2} or \frac{5 π}{6} < x < \frac{3 π}{2}

Apply the periodicity of

cos (x) - 2 cos (x) sin (x)

\frac{π}{6} + 2 πn < x < \frac{π}{2} + 2 πn or \frac{5 π}{6} + 2 πn < x < \frac{3 π}{2} + 2 πn

Popular Examples

7.5cos(pi/6 (x+3))+10.5>13.75

7.5 cos (\frac{π}{6} (x + 3)) + 10.5 > 13.75

sin(x^2)<0

sin (x^{2}) < 0

sin(x+pi/4)> 1/2

sin (x + \frac{π}{4}) > \frac{1}{2}

sin(x/3)>= sqrt(3/2)

sin (\frac{x}{3}) \geq \frac{3}{2}

2tan(x)<sqrt(2),0<= x<2pi

2 tan (x) < 2, 0 \leq x < 2 π