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Popular Trigonometry >

sin^4(x)-cos^4(x)<= 0

Solution

sin^{4} (x) - cos^{4} (x) \leq 0

Solution

πn \leq x \leq \frac{π}{4} + πn or \frac{3 π}{4} + πn \leq x \leq π + πn

Interval Notation

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

Decimal

πn \leq x \leq 0.78539 \dots + πn or 2.35619 \dots + πn \leq x \leq 3.14159 \dots + πn

Solution steps

sin^{4} (x) - cos^{4} (x) \leq 0

Periodicity of

sin^{4} (x) - cos^{4} (x) : π

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

sin^{4} (x), cos^{4} (x)

Periodicity of

sin^{4} (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

Periodicity of

cos^{4} (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

Combine periods:

π, π

= π

Factor

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

sin^{4} (x) - cos^{4} (x)

Rewrite

sin^{4} (x) - cos^{4} (x)

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

sin^{4} (x) - cos^{4} (x)

Apply exponent rule:

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

sin^{4} (x) = (sin^{2} (x))^{2}

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

Apply exponent rule:

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

cos^{4} (x) = (cos^{2} (x))^{2}

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

(sin^{2} (x) + cos^{2} (x)) (sin (x) + cos (x)) (sin (x) - cos (x)) \leq 0

To find the zeroes, set the inequality to zero

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

Solve

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

for

0 \leq x < π

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

Solving each part separately

sin (x) + cos (x) = 0 : x = \frac{3 π}{4}

sin (x) + cos (x) = 0, 0 \leq x < π

Rewrite using trig identities

sin (x) + cos (x) = 0

Divide both sides by

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

\frac{sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} + 1 = 0

Use the basic trigonometric identity:

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

tan (x) + 1 = 0

tan (x) + 1 = 0

Move

1

to the right side

tan (x) + 1 = 0

Subtract

1

from both sides

tan (x) + 1 - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Solutions for the range

0 \leq x < π

x = \frac{3 π}{4}

sin (x) - cos (x) = 0 : x = \frac{π}{4}

sin (x) - cos (x) = 0, 0 \leq x < π

Rewrite using trig identities

sin (x) - cos (x) = 0

Divide both sides by

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

\frac{sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} - 1 = 0

Use the basic trigonometric identity:

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

tan (x) - 1 = 0

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

tan (x) - 1 + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Solutions for the range

0 \leq x < π

x = \frac{π}{4}

Combine all the solutions

\frac{π}{4} or \frac{3 π}{4}

The intervals between the zeros

0 < x < \frac{π}{4}, \frac{π}{4} < x < \frac{3 π}{4}, \frac{3 π}{4} < x < π

Summarize in a table:

sin^{2} (x) + cos^{2} (x) sin (x) + cos (x) sin (x) - cos (x) (sin^{2} (x) + cos^{2} (x)) (sin (x) + cos (x)) (sin (x) - cos (x)) x = 0 + + - - 0 < x < \frac{π}{4} + + - - x = \frac{π}{4} + + 00 \frac{π}{4} < x < \frac{3 π}{4} + + + + x = \frac{3 π}{4} + 0 + 0 \frac{3 π}{4} < x < π + - + - x = π + - + -

Identify the intervals that satisfy the required condition:

\leq 0

x = 0 or 0 < x < \frac{π}{4} or x = \frac{π}{4} or x = \frac{3 π}{4} or \frac{3 π}{4} < x < π or x = π

Merge Overlapping Intervals

0 \leq x \leq \frac{π}{4} or \frac{3 π}{4} \leq x < π or x = π