What is the general solution for 8sin^2(x)+6cos^2(x)=10 ?

The general solution for 8sin^2(x)+6cos^2(x)=10 is No Solution for x\in\mathbb{R}

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Popular >

8sin^2(x)+6cos^2(x)=10

Solution

8 sin^{2} (x) + 6 cos^{2} (x) = 10

Solution

No Solution for x \in R

Solution steps

8 sin^{2} (x) + 6 cos^{2} (x) = 10

Subtract

10

from both sides

8 sin^{2} (x) + 6 cos^{2} (x) - 10 = 0

Rewrite using trig identities

- 10 + 6 cos^{2} (x) + 8 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 10 + 6 (1 - sin^{2} (x)) + 8 sin^{2} (x)

Simplify

- 10 + 6 (1 - sin^{2} (x)) + 8 sin^{2} (x) : 2 sin^{2} (x) - 4

- 10 + 6 (1 - sin^{2} (x)) + 8 sin^{2} (x)

Expand

6 (1 - sin^{2} (x)) : 6 - 6 sin^{2} (x)

6 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 6, b = 1, c = sin^{2} (x)

= 6 \cdot 1 - 6 sin^{2} (x)

Multiply the numbers:

6 \cdot 1 = 6

= 6 - 6 sin^{2} (x)

= - 10 + 6 - 6 sin^{2} (x) + 8 sin^{2} (x)

Simplify

- 10 + 6 - 6 sin^{2} (x) + 8 sin^{2} (x) : 2 sin^{2} (x) - 4

- 10 + 6 - 6 sin^{2} (x) + 8 sin^{2} (x)

Add similar elements:

- 6 sin^{2} (x) + 8 sin^{2} (x) = 2 sin^{2} (x)

= - 10 + 6 + 2 sin^{2} (x)

Add/Subtract the numbers:

- 10 + 6 = - 4

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

- 4 + 2 u^{2} = 0

- 4 + 2 u^{2} = 0 : u = 2, u = - 2

- 4 + 2 u^{2} = 0

Move

4

to the right side

- 4 + 2 u^{2} = 0

Add

4

to both sides

- 4 + 2 u^{2} + 4 = 0 + 4

Simplify

2 u^{2} = 4

2 u^{2} = 4

Divide both sides by

2

2 u^{2} = 4

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{4}{2}

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 8sin^2(x)+6cos^2(x)=10 ?
The general solution for 8sin^2(x)+6cos^2(x)=10 is No Solution for x\in\mathbb{R}