What is the general solution for 4sin^2(x)+2sin(x)-3=0 ?

The general solution for 4sin^2(x)+2sin(x)-3=0 is x=0.70941…+2pin,x=pi-0.70941…+2pin

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4sin^2(x)+2sin(x)-3=0

Solution

4 sin^{2} (x) + 2 sin (x) - 3 = 0

Solution

x = 0.70941 \dots + 2 πn, x = π - 0.70941 \dots + 2 πn

Degrees

x = 40.64631 \dots^{\circ} + 36 0^{\circ} n, x = 139.35368 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 sin^{2} (x) + 2 sin (x) - 3 = 0

Solve by substitution

4 sin^{2} (x) + 2 sin (x) - 3 = 0

Let:

sin (x) = u

4 u^{2} + 2 u - 3 = 0

4 u^{2} + 2 u - 3 = 0 : u = \frac{- 1 + 13}{4}, u = - \frac{1 + 13}{4}

4 u^{2} + 2 u - 3 = 0

Solve with the quadratic formula

4 u^{2} + 2 u - 3 = 0

Quadratic Equation Formula:

For

a = 4, b = 2, c = - 3

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 4 ( - 3 )}{2 \cdot 4}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 4 ( - 3 )}{2 \cdot 4}

2^{2} - 4 \cdot 4 (- 3) = 213

2^{2} - 4 \cdot 4 (- 3)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 2^{2} + 48

2^{2} = 4

= 4 + 48

Add the numbers:

4 + 48 = 52

= 52

Prime factorization of

52 : 2^{2} \cdot 13

52

52

divides by

252 = 26 \cdot 2

= 2 \cdot 26

26

divides by

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

Apply radical rule:

= 13 2^{2}

Apply radical rule:

2^{2} = 2

= 213

u_{1, 2} = \frac{- 2 \pm 2 13}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 2 + 2 13}{2 \cdot 4}, u_{2} = \frac{- 2 - 2 13}{2 \cdot 4}

u = \frac{- 2 + 2 13}{2 \cdot 4} : \frac{- 1 + 13}{4}

\frac{- 2 + 2 13}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 2 + 2 13}{8}

Factor

- 2 + 213 : 2 (- 1 + 13)

- 2 + 213

Rewrite as

= - 2 \cdot 1 + 213

Factor out common term

2

= 2 (- 1 + 13)

= \frac{2 ( - 1 + 13 )}{8}

Cancel the common factor:

2

= \frac{- 1 + 13}{4}

u = \frac{- 2 - 2 13}{2 \cdot 4} : - \frac{1 + 13}{4}

\frac{- 2 - 2 13}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 2 - 2 13}{8}

Factor

- 2 - 213 : - 2 (1 + 13)

- 2 - 213

Rewrite as

= - 2 \cdot 1 - 213

Factor out common term

2

= - 2 (1 + 13)

= - \frac{2 ( 1 + 13 )}{8}

Cancel the common factor:

2

= - \frac{1 + 13}{4}

The solutions to the quadratic equation are:

u = \frac{- 1 + 13}{4}, u = - \frac{1 + 13}{4}

Substitute back

u = sin (x)

sin (x) = \frac{- 1 + 13}{4}, sin (x) = - \frac{1 + 13}{4}

sin (x) = \frac{- 1 + 13}{4}, sin (x) = - \frac{1 + 13}{4}

sin (x) = \frac{- 1 + 13}{4} : x = arcsin (\frac{- 1 + 13}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 13}{4}) + 2 πn

sin (x) = \frac{- 1 + 13}{4}

Apply trig inverse properties

sin (x) = \frac{- 1 + 13}{4}

General solutions for

sin (x) = \frac{- 1 + 13}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{- 1 + 13}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 13}{4}) + 2 πn

x = arcsin (\frac{- 1 + 13}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 13}{4}) + 2 πn

sin (x) = - \frac{1 + 13}{4} :

No Solution

sin (x) = - \frac{1 + 13}{4}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{- 1 + 13}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 13}{4}) + 2 πn

Show solutions in decimal form

x = 0.70941 \dots + 2 πn, x = π - 0.70941 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x)+2sin(x)-3=0 ?
The general solution for 4sin^2(x)+2sin(x)-3=0 is x=0.70941…+2pin,x=pi-0.70941…+2pin