What is the general solution for cos^3(x)-sin^3(x)=0 ?

The general solution for cos^3(x)-sin^3(x)=0 is x= pi/4+pin

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Popular >

cos^3(x)-sin^3(x)=0

Solution

cos^{3} (x) - sin^{3} (x) = 0

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

cos^{3} (x) - sin^{3} (x) = 0

Factor

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

cos^{3} (x) - sin^{3} (x)

Apply Difference of Cubes Formula:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

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

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

Solving each part separately

cos (x) - sin (x) = 0 or cos (x) sin (x) + 1 = 0

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

cos (x) - sin (x) = 0

Rewrite using trig identities

cos (x) - sin (x) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (x) = 0

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

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

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 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

cos (x) sin (x) + 1 = 0 :

No Solution

cos (x) sin (x) + 1 = 0

Rewrite using trig identities

cos (x) sin (x) + 1

Use the Double Angle identity:

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

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

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

1 + \frac{sin ( 2 x )}{2} = 0

Move

1

to the right side

1 + \frac{sin ( 2 x )}{2} = 0

Subtract

1

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{4} + πn

Graph

Popular Examples

2sec(θ)+3=0

2 sec (θ) + 3 = 0

3sec^2(x)-5tan(x)=3

3 sec^{2} (x) - 5 tan (x) = 3

sin(2x)cos(6x)-cos(2x)sin(6x)=-0.55

sin (2 x) cos (6 x) - cos (2 x) sin (6 x) = - 0.55

1/(cos(x))=-1

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

3tan^3(θ)-tan(θ)=3tan^2(θ)-1

3 tan^{3} (θ) - tan (θ) = 3 tan^{2} (θ) - 1

Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)-sin^3(x)=0 ?
The general solution for cos^3(x)-sin^3(x)=0 is x= pi/4+pin