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Popular >

cos^6(x)+sin^6(x)>= 1/4

Solution

cos^{6} (x) + sin^{6} (x) \geq \frac{1}{4}

Solution

πn \leq x \leq π + πn

Interval Notation

[πn, π + πn]

Decimal Notation

πn \leq x \leq 3.14159 \dots + πn

Solution steps

cos^{6} (x) + sin^{6} (x) \geq \frac{1}{4}

Periodicity of

cos^{6} (x) + sin^{6} (x) : π

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

cos^{6} (x), sin^{6} (x)

Periodicity of

cos^{6} (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

sin^{6} (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

cos^{6} (x) + sin^{6} (x) : (cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x))

cos^{6} (x) + sin^{6} (x)

Rewrite

cos^{6} (x) + sin^{6} (x)

(cos^{2} (x))^{3} + (sin^{2} (x))^{3}

cos^{6} (x) + sin^{6} (x)

Apply exponent rule:

a^{b c} = (a^{b})^{c}

sin^{6} (x) = (sin^{2} (x))^{3}

= cos^{6} (x) + (sin^{2} (x))^{3}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

cos^{6} (x) = (cos^{2} (x))^{3}

= (cos^{2} (x))^{3} + (sin^{2} (x))^{3}

= (cos^{2} (x))^{3} + (sin^{2} (x))^{3}

Apply Sum of Cubes Formula:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

(cos^{2} (x))^{3} + (sin^{2} (x))^{3} = (cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x))

= (sin^{2} (x) + cos^{2} (x)) (sin^{4} (x) - sin^{2} (x) cos^{2} (x) + cos^{4} (x))

(cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x)) \geq \frac{1}{4}

To find the zeroes, set the inequality to zero

(cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x)) = 0

Solve

(cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x)) = 0

for

0 \leq x < π

(cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x)) = 0

Solving each part separately

The intervals between the zeros

0 < x < π

Summarize in a table:

cos^{2} (x) + sin^{2} (x) cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x) (cos^{2} (x) + sin^{2} (x)) (cos^{4} (x) - sin^{2} (x) cos^{2} (x) + sin^{4} (x)) x = 0 + + + 0 < x < π + + + x = π + + +

Identify the intervals that satisfy the required condition:

\geq 0

x = 0 or 0 < x < π or x = π

Merge Overlapping Intervals

0 \leq x < π or x = π

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

x = 0

0 < x < π

0 \leq x < π

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

0 \leq x < π

x = π

0 \leq x \leq π

0 \leq x \leq π

Apply the periodicity of

cos^{6} (x) + sin^{6} (x)

πn \leq x \leq π + πn

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3cos(t/2-pi/4)>0

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