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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

tan^2(θ)sin(θ)=-sin(θ)

Solution

tan^{2} (θ) sin (θ) = - sin (θ)

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

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

Solution steps

tan^{2} (θ) sin (θ) = - sin (θ)

Subtract

- sin (θ)

from both sides

tan^{2} (θ) sin (θ) + sin (θ) = 0

Factor

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

tan^{2} (θ) sin (θ) + sin (θ)

Factor out common term

sin (θ)

= sin (θ) (tan^{2} (θ) + 1)

sin (θ) (tan^{2} (θ) + 1) = 0

Solving each part separately

sin (θ) = 0 or tan^{2} (θ) + 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

tan^{2} (θ) + 1 = 0 :

No Solution

tan^{2} (θ) + 1 = 0

Solve by substitution

tan^{2} (θ) + 1 = 0

Let:

tan (θ) = 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 (θ)

tan (θ) = i, tan (θ) = - i

tan (θ) = i, tan (θ) = - i

tan (θ) = i :

No Solution

tan (θ) = i

No Solution

tan (θ) = - i :

No Solution

tan (θ) = - i

No Solution

Combine all the solutions

No Solution

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

Popular Examples

sec(x)=1+tan(x)

sec (x) = 1 + tan (x)

cos^2(x)=0,5

cos^{2} (x) = 0, 5

2cos(x)+3sin(x)=2

2 cos (x) + 3 sin (x) = 2

-cot(x)-1=csc(x)

- cot (x) - 1 = csc (x)

4sec(θ)+1=9

4 sec (θ) + 1 = 9

Frequently Asked Questions (FAQ)

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