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

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

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Popular >

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

Solution

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

Solution

y = πn

Degrees

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

Solution steps

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

Subtract

3 tan (y)

from both sides

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

Factor

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

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

Factor out common term

tan (y)

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

Factor

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

sin^{2} (y) - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (sin (y) + 3) (sin (y) - 3)

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

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

Solving each part separately

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

tan (y) = 0 : y = πn

tan (y) = 0

General solutions for

tan (y) = 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}

y = 0 + πn

y = 0 + πn

Solve

y = 0 + πn : y = πn

y = 0 + πn

0 + πn = πn

y = πn

y = πn

sin (y) + 3 = 0 :

No Solution

sin (y) + 3 = 0

Move

3

to the right side

sin (y) + 3 = 0

Subtract

3

from both sides

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

Simplify

sin (y) = - 3

sin (y) = - 3

- 1 \leq sin (x) \leq 1

No Solution

sin (y) - 3 = 0 :

No Solution

sin (y) - 3 = 0

Move

3

to the right side

sin (y) - 3 = 0

Add

3

to both sides

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

Simplify

sin (y) = 3

sin (y) = 3

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

y = πn

Popular Examples

4cos(x)-8sin(x)cos(x)=0

4 cos (x) - 8 sin (x) cos (x) = 0

sin(x)+cos(x)=a

sin (x) + cos (x) = a

sin(x)=-sqrt(1/2)

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

tan(x)-(sqrt(3))/3 =0

tan (x) - \frac{3}{3} = 0

tan(2θ)=0.5714

tan (2 θ) = 0.5714

Frequently Asked Questions (FAQ)

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