What is the general solution for sin^2(x)+3cos^2(x)=0 ?

The general solution for sin^2(x)+3cos^2(x)=0 is No Solution for x\in\mathbb{R}

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Popular >

sin^2(x)+3cos^2(x)=0

Solution

sin^{2} (x) + 3 cos^{2} (x) = 0

Solution

No Solution for x \in R

Solution steps

sin^{2} (x) + 3 cos^{2} (x) = 0

Rewrite using trig identities

sin^{2} (x) + 3 cos^{2} (x)

Use the Pythagorean identity:

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

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

= sin^{2} (x) + 3 (1 - sin^{2} (x))

Simplify

sin^{2} (x) + 3 (1 - sin^{2} (x)) : - 2 sin^{2} (x) + 3

sin^{2} (x) + 3 (1 - sin^{2} (x))

Expand

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

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

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

Simplify

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

sin^{2} (x) + 3 - 3 sin^{2} (x)

Group like terms

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

Add similar elements:

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

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

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

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

3 - 2 sin^{2} (x) = 0

Solve by substitution

3 - 2 sin^{2} (x) = 0

Let:

sin (x) = u

3 - 2 u^{2} = 0

3 - 2 u^{2} = 0 : u = \frac{3}{2}, u = - \frac{3}{2}

3 - 2 u^{2} = 0

Move

3

to the right side

3 - 2 u^{2} = 0

Subtract

3

from both sides

3 - 2 u^{2} - 3 = 0 - 3

Simplify

- 2 u^{2} = - 3

- 2 u^{2} = - 3

Divide both sides by

- 2

- 2 u^{2} = - 3

Divide both sides by

- 2

\frac{- 2 u ^{2}}{- 2} = \frac{- 3}{- 2}

Simplify

u^{2} = \frac{3}{2}

u^{2} = \frac{3}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3}{2}, u = - \frac{3}{2}

Substitute back

u = sin (x)

sin (x) = \frac{3}{2}, sin (x) = - \frac{3}{2}

sin (x) = \frac{3}{2}, sin (x) = - \frac{3}{2}

sin (x) = \frac{3}{2} :

No Solution

sin (x) = \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - \frac{3}{2} :

No Solution

sin (x) = - \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

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2 sin (x) cos (x) - 2 cos (x) - sin (x) + 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)+3cos^2(x)=0 ?
The general solution for sin^2(x)+3cos^2(x)=0 is No Solution for x\in\mathbb{R}