What is the general solution for tan^2(x)sin(x)=3sin(x) ?

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

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tan^2(x)sin(x)=3sin(x)

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = \frac{2 π}{3} + πn, x = \frac{π}{3} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

3 sin (x)

from both sides

tan^{2} (x) sin (x) - 3 sin (x) = 0

Factor

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

tan^{2} (x) sin (x) - 3 sin (x)

Factor out common term

sin (x)

= sin (x) (tan^{2} (x) - 3)

Factor

tan^{2} (x) - 3 : (tan (x) + 3) (tan (x) - 3)

tan^{2} (x) - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (tan (x) + 3) (tan (x) - 3)

= sin (x) (tan (x) + 3) (tan (x) - 3)

sin (x) (tan (x) + 3) (tan (x) - 3) = 0

Solving each part separately

sin (x) = 0 or tan (x) + 3 = 0 or tan (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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) + 3 = 0 : x = \frac{2 π}{3} + πn

tan (x) + 3 = 0

Move

3

to the right side

tan (x) + 3 = 0

Subtract

3

from both sides

tan (x) + 3 - 3 = 0 - 3

Simplify

tan (x) = - 3

tan (x) = - 3

General solutions for

tan (x) = - 3

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{2 π}{3} + πn

x = \frac{2 π}{3} + πn

tan (x) - 3 = 0 : x = \frac{π}{3} + πn

tan (x) - 3 = 0

Move

3

to the right side

tan (x) - 3 = 0

Add

3

to both sides

tan (x) - 3 + 3 = 0 + 3

Simplify

tan (x) = 3

tan (x) = 3

General solutions for

tan (x) = 3

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}{3} + πn

x = \frac{π}{3} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{2 π}{3} + πn, x = \frac{π}{3} + πn

Popular Examples

cos(θ)-sin(2θ)=0

cos (θ) - sin (2 θ) = 0

sin(7x)=0

sin (7 x) = 0

cos(2x)-sqrt(3)cos(x)=-1

cos (2 x) - 3 cos (x) = - 1

solvefor x,tan(x)=-1 solve for

x, tan (x) = - 1

2sin(x)=3cos(x)

2 sin (x) = 3 cos (x)

Frequently Asked Questions (FAQ)

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