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

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

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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn

Degrees

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

Solution steps

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

Subtract

- sin (x)

from both sides

tan^{2} (x) sin (x) + sin (x) = 0

Factor

tan^{2} (x) sin (x) + sin (x) : sin (x) (tan^{2} (x) + 1)

tan^{2} (x) sin (x) + sin (x)

Factor out common term

sin (x)

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

sin (x) (tan^{2} (x) + 1) = 0

Solving each part separately

sin (x) = 0 or tan^{2} (x) + 1 = 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^{2} (x) + 1 = 0 :

No Solution

tan^{2} (x) + 1 = 0

Solve by substitution

tan^{2} (x) + 1 = 0

Let:

tan (x) = u

u^{2} + 1 = 0

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

Move

1

to the right side

u^{2} + 1 = 0

Subtract

1

from both sides

u^{2} + 1 - 1 = 0 - 1

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

Substitute back

u = tan (x)

tan (x) = i, tan (x) = - i

tan (x) = i, tan (x) = - i

tan (x) = i :

No Solution

tan (x) = i

No Solution

tan (x) = - i :

No Solution

tan (x) = - i

No Solution

Combine all the solutions

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

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