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Popular >

2cos^2(X)<= 1-2cos(2X)

Solution

2 cos^{2} (X) \leq 1 - 2 cos (2 X)

Solution

0.95531 \dots + πn \leq X \leq - 0.95531 \dots + π + πn

Interval Notation

[0.95531 \dots + πn, - 0.95531 \dots + π + πn]

Decimal Notation

0.95531 \dots + πn \leq X \leq 2.18627 \dots + πn

Solution steps

2 cos^{2} (X) \leq 1 - 2 cos (2 X)

Move

2 cos (2 X)

to the left side

2 cos^{2} (X) \leq 1 - 2 cos (2 X)

Add

2 cos (2 X)

to both sides

2 cos^{2} (X) + 2 cos (2 X) \leq 1 - 2 cos (2 X) + 2 cos (2 X)

2 cos^{2} (X) + 2 cos (2 X) \leq 1

2 cos^{2} (X) + 2 cos (2 X) \leq 1

Use the following identity:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

2 (cos^{2} (X) - sin^{2} (X)) + 2 cos^{2} (X) \leq 1

Simplify

2 (cos^{2} (X) - sin^{2} (X)) + 2 cos^{2} (X) : 4 cos^{2} (X) - 2 sin^{2} (X)

2 (cos^{2} (X) - sin^{2} (X)) + 2 cos^{2} (X)

Expand

2 (cos^{2} (X) - sin^{2} (X)) : 2 cos^{2} (X) - 2 sin^{2} (X)

2 (cos^{2} (X) - sin^{2} (X))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = cos^{2} (X), c = sin^{2} (X)

= 2 cos^{2} (X) - 2 sin^{2} (X)

= 2 cos^{2} (X) - 2 sin^{2} (X) + 2 cos^{2} (X)

Add similar elements:

2 cos^{2} (X) + 2 cos^{2} (X) = 4 cos^{2} (X)

= 4 cos^{2} (X) - 2 sin^{2} (X)

4 cos^{2} (X) - 2 sin^{2} (X) \leq 1

Periodicity of

4 cos^{2} (X) - 2 sin^{2} (X) : π

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

4 cos^{2} (X), 2 sin^{2} (X)

Periodicity of

4 cos^{2} (X) : π

Periodicity of

cos^{n} (x) = \frac{Periodicity of cos ( x )}{2},

if n is even

Periodicity of

cos (X) : 2 π

Periodicity of

cos (x)

2 π

= 2 π

\frac{2 π}{2}

Simplify

π

Periodicity of

2 sin^{2} (X) : π

Periodicity of

sin^{n} (x) = \frac{Periodicity of sin ( x )}{2},

if n is even

Periodicity of

sin (X) : 2 π

Periodicity of

sin (x)

2 π

= 2 π

\frac{2 π}{2}

Simplify

π

Combine periods:

π, π

= π

Factor

4 cos^{2} (X) - 2 sin^{2} (X) : 2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X))

4 cos^{2} (X) - 2 sin^{2} (X)

Rewrite

4

2 \cdot 2

= 2 \cdot 2 cos^{2} (X) - 2 sin^{2} (X)

Factor out common term

2

= 2 (2 cos^{2} (X) - sin^{2} (X))

Factor

2 cos^{2} (X) - sin^{2} (X) : (2 cos (X) + sin (X)) (2 cos (X) - sin (X))

2 cos^{2} (X) - sin^{2} (X)

Rewrite

2 cos^{2} (X) - sin^{2} (X)

(2 cos (X))^{2} - sin^{2} (X)

2 cos^{2} (X) - sin^{2} (X)

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (X) - sin^{2} (X)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} cos^{2} (X) = (2 cos (X))^{2}

= (2 cos (X))^{2} - sin^{2} (X)

= (2 cos (X))^{2} - sin^{2} (X)

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 cos (X))^{2} - sin^{2} (X) = (2 cos (X) + sin (X)) (2 cos (X) - sin (X))

= (2 cos (X) + sin (X)) (2 cos (X) - sin (X))

= 2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X))

2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X)) \leq 1

To find the zeroes, set the inequality to zero

2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X)) = 0

Solve

2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X)) = 0

for

0 \leq X < π

2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X)) = 0

Solving each part separately

2 cos (X) + sin (X) = 0 : X = - 0.95531 \dots + π

2 cos (X) + sin (X) = 0, 0 \leq X < π

Rewrite using trig identities

2 cos (X) + sin (X) = 0

Divide both sides by

cos (X), cos (X) \neq = 0

\frac{2 cos ( X ) + sin ( X )}{cos ( X )} = \frac{0}{cos ( X )}

Simplify

2 + \frac{sin ( X )}{cos ( X )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

2 + tan (X) = 0

2 + tan (X) = 0

Move

2

to the right side

2 + tan (X) = 0

Subtract

2

from both sides

2 + tan (X) - 2 = 0 - 2

Simplify

tan (X) = - 2

tan (X) = - 2

Apply trig inverse properties

tan (X) = - 2

General solutions for

tan (X) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

X = arctan (- 2) + πn

X = arctan (- 2) + πn

Solutions for the range

0 \leq X < π

X = - arctan (2) + π

Show solutions in decimal form

X = - 0.95531 \dots + π

2 cos (X) - sin (X) = 0 : X = 0.95531 \dots

2 cos (X) - sin (X) = 0, 0 \leq X < π

Rewrite using trig identities

2 cos (X) - sin (X) = 0

Divide both sides by

cos (X), cos (X) \neq = 0

\frac{2 cos ( X ) - sin ( X )}{cos ( X )} = \frac{0}{cos ( X )}

Simplify

2 - \frac{sin ( X )}{cos ( X )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

2 - tan (X) = 0

2 - tan (X) = 0

Move

2

to the right side

2 - tan (X) = 0

Subtract

2

from both sides

2 - tan (X) - 2 = 0 - 2

Simplify

- tan (X) = - 2

- tan (X) = - 2

Divide both sides by

- 1

- tan (X) = - 2

Divide both sides by

- 1

\frac{- tan ( X )}{- 1} = \frac{- 2}{- 1}

Simplify

tan (X) = 2

tan (X) = 2

Apply trig inverse properties

tan (X) = 2

General solutions for

tan (X) = 2

tan (x) = a \Rightarrow x = arctan (a) + πn

X = arctan (2) + πn

X = arctan (2) + πn

Solutions for the range

0 \leq X < π

X = arctan (2)

Show solutions in decimal form

X = 0.95531 \dots

Combine all the solutions

0.95531 \dots or - 0.95531 \dots + π

The intervals between the zeros

0 < X < 0.95531 \dots, 0.95531 \dots < X < - 0.95531 \dots + π, - 0.95531 \dots + π < X < π

Summarize in a table:

2 cos (X) + sin (X) 2 cos (X) - sin (X) 2 (2 cos (X) + sin (X)) (2 cos (X) - sin (X)) X = 0 + + + 0 < X < 0.95531 \dots + + + X = 0.95531 \dots + 00 0.95531 \dots < X < - 0.95531 \dots + π + - - X = - 0.95531 \dots + π 0 - 0 - 0.95531 \dots + π < X < π - - + X = π - - +

Identify the intervals that satisfy the required condition:

\leq 0

X = 0.95531 \dots or 0.95531 \dots < X < - 0.95531 \dots + π or X = - 0.95531 \dots + π

Merge Overlapping Intervals

0.95531 \dots \leq X < - 0.95531 \dots + π or X = - 0.95531 \dots + π

The union of two intervals is the set of numbers which are in either interval

X = 0.95531 \dots

0.95531 \dots < X < - 0.95531 \dots + π

0.95531 \dots \leq X < - 0.95531 \dots + π

The union of two intervals is the set of numbers which are in either interval

0.95531 \dots \leq X < - 0.95531 \dots + π

X = - 0.95531 \dots + π

0.95531 \dots \leq X \leq - 0.95531 \dots + π

0.95531 \dots \leq X \leq - 0.95531 \dots + π

Apply the periodicity of

4 cos^{2} (X) - 2 sin^{2} (X)

0.95531 \dots + πn \leq X \leq - 0.95531 \dots + π + πn

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