What is the general solution for 2sin^2(x)+4cos^2(x)=1 ?

The general solution for 2sin^2(x)+4cos^2(x)=1 is No Solution for x\in\mathbb{R}

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Popular >

2sin^2(x)+4cos^2(x)=1

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

1

from both sides

2 sin^{2} (x) + 4 cos^{2} (x) - 1 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

- 1 + 2 sin^{2} (x) + 4 (1 - sin^{2} (x)) : - 2 sin^{2} (x) + 3

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

4 \cdot 1 = 4

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

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

Simplify

- 1 + 2 sin^{2} (x) + 4 - 4 sin^{2} (x) : - 2 sin^{2} (x) + 3

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

Group like terms

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

Add similar elements:

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

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

Add/Subtract the numbers:

- 1 + 4 = 3

= - 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

Popular Examples

-sin(θ)= n/2-3

- sin (θ) = \frac{n}{2} - 3

3sin(θ)+2sin^2(θ)=-1,0<= θ<2pi

3 sin (θ) + 2 sin^{2} (θ) = - 1, 0 \leq θ < 2 π

3sin^2(x)+4sin(x)+1=0

3 sin^{2} (x) + 4 sin (x) + 1 = 0

cos(x)= 9/14

cos (x) = \frac{9}{14}

sin(θ)= 7/9

sin (θ) = \frac{7}{9}

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)+4cos^2(x)=1 ?
The general solution for 2sin^2(x)+4cos^2(x)=1 is No Solution for x\in\mathbb{R}