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

The general solution for tan(x)sin^2(x)=3tan(x) is x=pin

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Popular >

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

Solution

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

Solution

x = πn

Degrees

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

Solution steps

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

Subtract

3 tan (x)

from both sides

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

Factor

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

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

Factor out common term

tan (x)

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

Factor

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

sin^{2} (x) - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply Difference of Two Squares Formula:

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

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

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

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

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

Solving each part separately

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

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

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 = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sin (x) + 3 = 0 :

No Solution

sin (x) + 3 = 0

Move

3

to the right side

sin (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

sin (x) = - 3

sin (x) = - 3

- 1 \leq sin (x) \leq 1

No Solution

sin (x) - 3 = 0 :

No Solution

sin (x) - 3 = 0

Move

3

to the right side

sin (x) - 3 = 0

Add

3

to both sides

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

Simplify

sin (x) = 3

sin (x) = 3

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = πn

Popular Examples

tan^2(x)-sqrt(3)tan(x)=0

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

sin(x)=tan(x)

sin (x) = tan (x)

sin(3x)=-1/2

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

cos(x/2)= 1/2

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

csc(x)+cot(x)=1,0<= x<= 2pi

csc (x) + cot (x) = 1, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

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