What is the general solution for 2sin^2(θ)+5sin(θ)-12=0 ?

The general solution for 2sin^2(θ)+5sin(θ)-12=0 is No Solution for θ\in\mathbb{R}

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Popular >

2sin^2(θ)+5sin(θ)-12=0

Solution

2 sin^{2} (θ) + 5 sin (θ) - 12 = 0

Solution

No Solution for θ \in R

Solution steps

2 sin^{2} (θ) + 5 sin (θ) - 12 = 0

Solve by substitution

2 sin^{2} (θ) + 5 sin (θ) - 12 = 0

Let:

sin (θ) = u

2 u^{2} + 5 u - 12 = 0

2 u^{2} + 5 u - 12 = 0 : u = \frac{3}{2}, u = - 4

2 u^{2} + 5 u - 12 = 0

Solve with the quadratic formula

2 u^{2} + 5 u - 12 = 0

Quadratic Equation Formula:

For

a = 2, b = 5, c = - 12

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 ( - 12 )}{2 \cdot 2}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 ( - 12 )}{2 \cdot 2}

5^{2} - 4 \cdot 2 (- 12) = 11

5^{2} - 4 \cdot 2 (- 12)

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 2 \cdot 12

Multiply the numbers:

4 \cdot 2 \cdot 12 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

Add the numbers:

25 + 96 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- 5 \pm 11}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 5 + 11}{2 \cdot 2}, u_{2} = \frac{- 5 - 11}{2 \cdot 2}

u = \frac{- 5 + 11}{2 \cdot 2} : \frac{3}{2}

\frac{- 5 + 11}{2 \cdot 2}

Add/Subtract the numbers:

- 5 + 11 = 6

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{6}{4}

Cancel the common factor:

2

= \frac{3}{2}

u = \frac{- 5 - 11}{2 \cdot 2} : - 4

\frac{- 5 - 11}{2 \cdot 2}

Subtract the numbers:

- 5 - 11 = - 16

= \frac{- 16}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 16}{4}

Apply the fraction rule:

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

= - \frac{16}{4}

Divide the numbers:

\frac{16}{4} = 4

= - 4

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - 4

Substitute back

u = sin (θ)

sin (θ) = \frac{3}{2}, sin (θ) = - 4

sin (θ) = \frac{3}{2}, sin (θ) = - 4

sin (θ) = \frac{3}{2} :

No Solution

sin (θ) = \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

sin (θ) = - 4 :

No Solution

sin (θ) = - 4

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for θ \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(θ)+5sin(θ)-12=0 ?
The general solution for 2sin^2(θ)+5sin(θ)-12=0 is No Solution for θ\in\mathbb{R}