What is the general solution for 2sin^2(x)+7sin(x)-4=0 ?

The general solution for 2sin^2(x)+7sin(x)-4=0 is x= pi/6+2pin,x=(5pi)/6+2pin

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2sin^2(x)+7sin(x)-4=0

Solution

2 sin^{2} (x) + 7 sin (x) - 4 = 0

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin^{2} (x) + 7 sin (x) - 4 = 0

Solve by substitution

2 sin^{2} (x) + 7 sin (x) - 4 = 0

Let:

sin (x) = u

2 u^{2} + 7 u - 4 = 0

2 u^{2} + 7 u - 4 = 0 : u = \frac{1}{2}, u = - 4

2 u^{2} + 7 u - 4 = 0

Solve with the quadratic formula

2 u^{2} + 7 u - 4 = 0

Quadratic Equation Formula:

For

a = 2, b = 7, c = - 4

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

7^{2} - 4 \cdot 2 (- 4) = 9

7^{2} - 4 \cdot 2 (- 4)

Apply rule

- (- a) = a

= 7^{2} + 4 \cdot 2 \cdot 4

Multiply the numbers:

4 \cdot 2 \cdot 4 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

Add the numbers:

49 + 32 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- 7 \pm 9}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 7 + 9}{2 \cdot 2}, u_{2} = \frac{- 7 - 9}{2 \cdot 2}

u = \frac{- 7 + 9}{2 \cdot 2} : \frac{1}{2}

\frac{- 7 + 9}{2 \cdot 2}

Add/Subtract the numbers:

- 7 + 9 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- 7 - 9}{2 \cdot 2} : - 4

\frac{- 7 - 9}{2 \cdot 2}

Subtract the numbers:

- 7 - 9 = - 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{1}{2}, u = - 4

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - 4

sin (x) = \frac{1}{2}, sin (x) = - 4

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - 4 :

No Solution

sin (x) = - 4

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)+7sin(x)-4=0 ?
The general solution for 2sin^2(x)+7sin(x)-4=0 is x= pi/6+2pin,x=(5pi)/6+2pin