What is the general solution for sin^2(x)+2cos^2(x)=0 ?

The general solution for sin^2(x)+2cos^2(x)=0 is No Solution for x\in\mathbb{R}

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Popular >

sin^2(x)+2cos^2(x)=0

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

2 (1 - sin^{2} (x)) : 2 - 2 sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

= 2 \cdot 1 - 2 sin^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

2 - u^{2} = 0

2 - u^{2} = 0 : u = 2, u = - 2

2 - u^{2} = 0

Move

2

to the right side

2 - u^{2} = 0

Subtract

2

from both sides

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

Simplify

- u^{2} = - 2

- u^{2} = - 2

Divide both sides by

- 1

- u^{2} = - 2

Divide both sides by

- 1

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

Simplify

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

Substitute back

u = sin (x)

sin (x) = 2, sin (x) = - 2

sin (x) = 2, sin (x) = - 2

sin (x) = 2 :

No Solution

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - 2 :

No Solution

sin (x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)+2cos^2(x)=0 ?
The general solution for sin^2(x)+2cos^2(x)=0 is No Solution for x\in\mathbb{R}