What is the general solution for cos^2(x)-4sin^2(x)=2 ?

The general solution for cos^2(x)-4sin^2(x)=2 is No Solution for x\in\mathbb{R}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(x)-4sin^2(x)=2

Solution

cos^{2} (x) - 4 sin^{2} (x) = 2

Solution

No Solution for x \in R

Solution steps

cos^{2} (x) - 4 sin^{2} (x) = 2

Subtract

2

from both sides

cos^{2} (x) - 4 sin^{2} (x) - 2 = 0

Rewrite using trig identities

- 2 + cos^{2} (x) - 4 sin^{2} (x)

Use the Pythagorean identity:

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

cos^{2} (x) = 1 - sin^{2} (x)

= - 2 + 1 - sin^{2} (x) - 4 sin^{2} (x)

Simplify

- 2 + 1 - sin^{2} (x) - 4 sin^{2} (x) : - 5 sin^{2} (x) - 1

- 2 + 1 - sin^{2} (x) - 4 sin^{2} (x)

Add similar elements:

- sin^{2} (x) - 4 sin^{2} (x) = - 5 sin^{2} (x)

= - 2 + 1 - 5 sin^{2} (x)

Add/Subtract the numbers:

- 2 + 1 = - 1

= - 5 sin^{2} (x) - 1

= - 5 sin^{2} (x) - 1

- 1 - 5 sin^{2} (x) = 0

Solve by substitution

- 1 - 5 sin^{2} (x) = 0

Let:

sin (x) = u

- 1 - 5 u^{2} = 0

- 1 - 5 u^{2} = 0 : u = i \frac{1}{5}, u = - i \frac{1}{5}

- 1 - 5 u^{2} = 0

Move

1

to the right side

- 1 - 5 u^{2} = 0

Add

1

to both sides

- 1 - 5 u^{2} + 1 = 0 + 1

Simplify

- 5 u^{2} = 1

- 5 u^{2} = 1

Divide both sides by

- 5

- 5 u^{2} = 1

Divide both sides by

- 5

\frac{- 5 u ^{2}}{- 5} = \frac{1}{- 5}

Simplify

u^{2} = - \frac{1}{5}

u^{2} = - \frac{1}{5}

For

x^{2} = f (a)

the solutions are

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

u = - \frac{1}{5}, u = - - \frac{1}{5}

Simplify

- \frac{1}{5} : i \frac{1}{5}

- \frac{1}{5}

Apply radical rule:

- a = - 1 a

- \frac{1}{5} = - 1 \frac{1}{5}

= - 1 \frac{1}{5}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{5}

Simplify

- - \frac{1}{5} : - i \frac{1}{5}

- - \frac{1}{5}

Simplify

- \frac{1}{5} : i \frac{1}{5}

- \frac{1}{5}

Apply radical rule:

- a = - 1 a

- \frac{1}{5} = - 1 \frac{1}{5}

= - 1 \frac{1}{5}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{5}

= - i \frac{1}{5}

u = i \frac{1}{5}, u = - i \frac{1}{5}

Substitute back

u = sin (x)

sin (x) = i \frac{1}{5}, sin (x) = - i \frac{1}{5}

sin (x) = i \frac{1}{5}, sin (x) = - i \frac{1}{5}

sin (x) = i \frac{1}{5} :

No Solution

sin (x) = i \frac{1}{5}

No Solution

sin (x) = - i \frac{1}{5} :

No Solution

sin (x) = - i \frac{1}{5}

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

sin(x)=0.53

sin (x) = 0.53

2cos(2θ)=sqrt(2)

2 cos (2 θ) = 2

2+sin(x)=0

2 + sin (x) = 0

sin(x)=0.45

sin (x) = 0.45

sin(x)=0.42

sin (x) = 0.42

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)-4sin^2(x)=2 ?
The general solution for cos^2(x)-4sin^2(x)=2 is No Solution for x\in\mathbb{R}