What is the general solution for sin^2(θ)-sin(θ)-12=0 ?

The general solution for sin^2(θ)-sin(θ)-12=0 is No Solution for θ\in\mathbb{R}

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Popular >

sin^2(θ)-sin(θ)-12=0

Solution

sin^{2} (θ) - sin (θ) - 12 = 0

Solution

No Solution for θ \in R

Solution steps

sin^{2} (θ) - sin (θ) - 12 = 0

Solve by substitution

sin^{2} (θ) - sin (θ) - 12 = 0

Let:

sin (θ) = u

u^{2} - u - 12 = 0

u^{2} - u - 12 = 0 : u = 4, u = - 3

u^{2} - u - 12 = 0

Solve with the quadratic formula

u^{2} - u - 12 = 0

Quadratic Equation Formula:

For

a = 1, b = - 1, c = - 12

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot ( - 12 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot ( - 12 )}{2 \cdot 1}

(- 1)^{2} - 4 \cdot 1 \cdot (- 12) = 7

(- 1)^{2} - 4 \cdot 1 \cdot (- 12)

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 1 \cdot 12

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 1 \cdot 12 = 48

4 \cdot 1 \cdot 12

Multiply the numbers:

4 \cdot 1 \cdot 12 = 48

= 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 7}{2 \cdot 1}, u_{2} = \frac{- ( - 1 ) - 7}{2 \cdot 1}

u = \frac{- ( - 1 ) + 7}{2 \cdot 1} : 4

\frac{- ( - 1 ) + 7}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{1 + 7}{2 \cdot 1}

Add the numbers:

1 + 7 = 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 ) - 7}{2 \cdot 1} : - 3

\frac{- ( - 1 ) - 7}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{1 - 7}{2 \cdot 1}

Subtract the numbers:

1 - 7 = - 6

= \frac{- 6}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6}{2}

Apply the fraction rule:

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

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

The solutions to the quadratic equation are:

u = 4, u = - 3

Substitute back

u = sin (θ)

sin (θ) = 4, sin (θ) = - 3

sin (θ) = 4, sin (θ) = - 3

sin (θ) = 4 :

No Solution

sin (θ) = 4

- 1 \leq sin (x) \leq 1

No Solution

sin (θ) = - 3 :

No Solution

sin (θ) = - 3

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(θ)-sin(θ)-12=0 ?
The general solution for sin^2(θ)-sin(θ)-12=0 is No Solution for θ\in\mathbb{R}