What is the general solution for sin^3(x)=sin^3(x)tan(x) ?

The general solution for sin^3(x)=sin^3(x)tan(x) is x=2pin,x=pi+2pin,x= pi/4+pin

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sin^3(x)=sin^3(x)tan(x)

sin^{3} (x) = sin^{3} (x) tan (x)

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sin^{3} (x) = sin^{3} (x) tan (x)

Subtract

sin^{3} (x) tan (x)

from both sides

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

Factor

sin^{3} (x) - sin^{3} (x) tan (x) : - sin^{3} (x) (tan (x) - 1)

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

Factor out common term

- sin^{3} (x)

= - sin^{3} (x) (- 1 + tan (x))

- sin^{3} (x) (tan (x) - 1) = 0

Solving each part separately

sin^{3} (x) = 0 or tan (x) - 1 = 0

sin^{3} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{3} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

tan (x) - 1 = 0 : x = \frac{π}{4} + πn

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

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

- 4 sin^{2} (θ) - 7 sin (θ) + 7 = 0

2 cos^{2} (4 x) = 1

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

csc^{2} (x) + 3 csc (x) + 2 = 0

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

What is the general solution for sin^3(x)=sin^3(x)tan(x) ?
The general solution for sin^3(x)=sin^3(x)tan(x) is x=2pin,x=pi+2pin,x= pi/4+pin