What is the general solution for sin^2(x)=20-sin(x) ?

The general solution for sin^2(x)=20-sin(x) is No Solution for x\in\mathbb{R}

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Popular >

sin^2(x)=20-sin(x)

Solution

sin^{2} (x) = 20 - sin (x)

Solution

No Solution for x \in R

Solution steps

sin^{2} (x) = 20 - sin (x)

Solve by substitution

sin^{2} (x) = 20 - sin (x)

Let:

sin (x) = u

u^{2} = 20 - u

u^{2} = 20 - u : u = 4, u = - 5

u^{2} = 20 - u

Move

u

to the left side

u^{2} = 20 - u

Add

u

to both sides

u^{2} + u = 20 - u + u

Simplify

u^{2} + u = 20

u^{2} + u = 20

Move

20

to the left side

u^{2} + u = 20

Subtract

20

from both sides

u^{2} + u - 20 = 20 - 20

Simplify

u^{2} + u - 20 = 0

u^{2} + u - 20 = 0

Solve with the quadratic formula

u^{2} + u - 20 = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 20

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 20 )}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 20 )}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot (- 20) = 9

1^{2} - 4 \cdot 1 \cdot (- 20)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 20)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 20

Multiply the numbers:

4 \cdot 1 \cdot 20 = 80

= 1 + 80

Add the numbers:

1 + 80 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 1 \pm 9}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 1 + 9}{2 \cdot 1}, u_{2} = \frac{- 1 - 9}{2 \cdot 1}

u = \frac{- 1 + 9}{2 \cdot 1} : 4

\frac{- 1 + 9}{2 \cdot 1}

Add/Subtract the numbers:

- 1 + 9 = 8

= \frac{8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

u = \frac{- 1 - 9}{2 \cdot 1} : - 5

\frac{- 1 - 9}{2 \cdot 1}

Subtract the numbers:

- 1 - 9 = - 10

= \frac{- 10}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 10}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{10}{2}

Divide the numbers:

\frac{10}{2} = 5

= - 5

The solutions to the quadratic equation are:

u = 4, u = - 5

Substitute back

u = sin (x)

sin (x) = 4, sin (x) = - 5

sin (x) = 4, sin (x) = - 5

sin (x) = 4 :

No Solution

sin (x) = 4

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - 5 :

No Solution

sin (x) = - 5

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)=20-sin(x) ?
The general solution for sin^2(x)=20-sin(x) is No Solution for x\in\mathbb{R}