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4sin(2x)+3cos(x)<= 1

Solution

4 sin (2 x) + 3 cos (x) \leq 1

Solution

\frac{π}{2} + 2 πn \leq x \leq π + 0.38439 \dots + 2 πn or \frac{3 π}{2} + 2 πn \leq x \leq - 0.38439 \dots + 2 π + 2 πn

Interval Notation

[\frac{π}{2} + 2 πn, π + 0.38439 \dots + 2 πn] \cup [\frac{3 π}{2} + 2 πn, - 0.38439 \dots + 2 π + 2 πn]

Decimal Notation

1.57079 \dots + 2 πn \leq x \leq 3.52598 \dots + 2 πn or 4.71238 \dots + 2 πn \leq x \leq 5.89878 \dots + 2 πn

Solution steps

4 sin (2 x) + 3 cos (x) \leq 1

Use the following identity:

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

3 cos (x) + 4 \cdot 2 cos (x) sin (x) \leq 1

Simplify

3 cos (x) + 8 cos (x) sin (x) \leq 1

Periodicity of

3 cos (x) + 8 cos (x) sin (x) : 2 π

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

3 cos (x), 8 cos (x) sin (x)

Periodicity of

3 cos (x) : 2 π

Periodicity of

a \cdot cos (b x + c) + d = \frac{periodicity of cos ( x )}{∣ b ∣}

Periodicity of

cos (x)

2 π

= \frac{2 π}{∣ 1 ∣}

Simplify

= 2 π

Periodicity of

8 cos (x) sin (x) : π

8 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

3 cos (x) + 8 cos (x) sin (x) : cos (x) (8 sin (x) + 3)

3 cos (x) + 8 cos (x) sin (x)

Factor out common term

cos (x)

= cos (x) (3 + 8 sin (x))

cos (x) (8 sin (x) + 3) \leq 1

To find the zeroes, set the inequality to zero

cos (x) (8 sin (x) + 3) = 0

Solve

cos (x) (8 sin (x) + 3) = 0

for

0 \leq x < 2 π

cos (x) (8 sin (x) + 3) = 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}

8 sin (x) + 3 = 0 : x = π + 0.38439 \dots or x = - 0.38439 \dots + 2 π

8 sin (x) + 3 = 0, 0 \leq x < 2 π

Move

3

to the right side

8 sin (x) + 3 = 0

Subtract

3

from both sides

8 sin (x) + 3 - 3 = 0 - 3

Simplify

8 sin (x) = - 3

8 sin (x) = - 3

Divide both sides by

8

8 sin (x) = - 3

Divide both sides by

8

\frac{8 sin ( x )}{8} = \frac{- 3}{8}

Simplify

sin (x) = - \frac{3}{8}

sin (x) = - \frac{3}{8}

Apply trig inverse properties

sin (x) = - \frac{3}{8}

General solutions for

sin (x) = - \frac{3}{8}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{3}{8}) + 2 πn, x = π + arcsin (\frac{3}{8}) + 2 πn

x = arcsin (- \frac{3}{8}) + 2 πn, x = π + arcsin (\frac{3}{8}) + 2 πn

Solutions for the range

0 \leq x < 2 π

x = π + arcsin (\frac{3}{8}), x = - arcsin (\frac{3}{8}) + 2 π

Show solutions in decimal form

x = π + 0.38439 \dots, x = - 0.38439 \dots + 2 π

Combine all the solutions

\frac{π}{2} or π + 0.38439 \dots or \frac{3 π}{2} or - 0.38439 \dots + 2 π

The intervals between the zeros

0 < x < \frac{π}{2}, \frac{π}{2} < x < π + 0.38439 \dots, π + 0.38439 \dots < x < \frac{3 π}{2}, \frac{3 π}{2} < x < - 0.38439 \dots + 2 π, - 0.38439 \dots + 2 π < x < 2 π

Summarize in a table:

cos (x) 8 sin (x) + 3 cos (x) (8 sin (x) + 3) x = 0 + + + 0 < x < \frac{π}{2} + + + x = \frac{π}{2} 0 + 0 \frac{π}{2} < x < π + 0.38439 \dots - + - x = π + 0.38439 \dots - 00 π + 0.38439 \dots < x < \frac{3 π}{2} - - + x = \frac{3 π}{2} 0 - 0 \frac{3 π}{2} < x < - 0.38439 \dots + 2 π + - - x = - 0.38439 \dots + 2 π + 00 - 0.38439 \dots + 2 π < x < 2 π + + + x = 2 π + + +

Identify the intervals that satisfy the required condition:

\leq 0

x = \frac{π}{2} or \frac{π}{2} < x < π + 0.38439 \dots or x = π + 0.38439 \dots or x = \frac{3 π}{2} or \frac{3 π}{2} < x < - 0.38439 \dots + 2 π or x = - 0.38439 \dots + 2 π

Merge Overlapping Intervals

\frac{π}{2} \leq x \leq π + 0.38439 \dots or \frac{3 π}{2} \leq x < - 0.38439 \dots + 2 π or x = - 0.38439 \dots + 2 π