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

The general solution for 2sin^2(x)+(sqrt(3)-4)sin(x)=2sqrt(3) is x=(4pi)/3+2pin,x=(5pi)/3+2pin

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Popular >

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

Solution

2 sin^{2} (x) + (3 - 4) sin (x) = 23

Solution

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin^{2} (x) + (3 - 4) sin (x) = 23

Solve by substitution

2 sin^{2} (x) + (3 - 4) sin (x) = 23

Let:

sin (x) = u

2 u^{2} + (3 - 4) u = 23

2 u^{2} + (3 - 4) u = 23 : u = 2, u = - \frac{3}{2}

2 u^{2} + (3 - 4) u = 23

Move

23

to the left side

2 u^{2} + (3 - 4) u = 23

Subtract

23

from both sides

2 u^{2} + (3 - 4) u - 23 = 23 - 23

Simplify

2 u^{2} + (3 - 4) u - 23 = 0

2 u^{2} + (3 - 4) u - 23 = 0

Solve with the quadratic formula

2 u^{2} + (3 - 4) u - 23 = 0

Quadratic Equation Formula:

For

a = 2, b = 3 - 4, c = - 23

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

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

(3 - 4)^{2} - 4 \cdot 2 (- 23) = 3 + 4

(3 - 4)^{2} - 4 \cdot 2 (- 23)

Apply rule

- (- a) = a

= (3 - 4)^{2} + 4 \cdot 2 \cdot 23

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= (3 - 4)^{2} + 163

Expand

(3 - 4)^{2} + 163 : 19 + 83

(3 - 4)^{2} + 163

(3 - 4)^{2} : 19 - 83

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 3, b = 4

= (3)^{2} - 23 \cdot 4 + 4^{2}

Simplify

(3)^{2} - 23 \cdot 4 + 4^{2} : 19 - 83

(3)^{2} - 23 \cdot 4 + 4^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 3

23 \cdot 4 = 83

23 \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 83

4^{2} = 16

4^{2}

4^{2} = 16

= 16

= 3 - 83 + 16

Add the numbers:

3 + 16 = 19

= 19 - 83

= 19 - 83

= 19 - 83 + 163

Add similar elements:

- 83 + 163 = 83

= 19 + 83

= 19 + 83

= 3 + 83 + 16

= (3)^{2} + 83 + (16)^{2}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= (3)^{2} + 83 + 4^{2}

23 \cdot 4 = 83

23 \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 83

= (3)^{2} + 23 \cdot 4 + 4^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(3)^{2} + 23 \cdot 4 + 4^{2} = (3 + 4)^{2}

= (3 + 4)^{2}

Apply radical rule:

(3 + 4)^{2} = 3 + 4

= 3 + 4

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

Separate the solutions

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

u = \frac{- ( 3 - 4 ) + 3 + 4}{2 \cdot 2} : 2

\frac{- ( 3 - 4 ) + 3 + 4}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( 3 - 4 ) + 3 + 4}{4}

Expand

- (3 - 4) + 3 + 4 : 8

- (3 - 4) + 3 + 4

- (3 - 4) : - 3 + 4

- (3 - 4)

Distribute parentheses

= - (3) - (- 4)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 3 + 4

= - 3 + 4 + 3 + 4

Simplify

- 3 + 4 + 3 + 4 : 8

- 3 + 4 + 3 + 4

Add similar elements:

- 3 + 3 = 0

= 4 + 4

Add the numbers:

4 + 4 = 8

= 8

= 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

u = \frac{- ( 3 - 4 ) - ( 3 + 4 )}{2 \cdot 2} : - \frac{3}{2}

\frac{- ( 3 - 4 ) - ( 3 + 4 )}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( 3 - 4 ) - ( 4 + 3 )}{4}

Expand

- (3 - 4) - (3 + 4) : - 23

- (3 - 4) - (3 + 4)

- (3 - 4) : - 3 + 4

- (3 - 4)

Distribute parentheses

= - (3) - (- 4)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 3 + 4

= - 3 + 4 - (3 + 4)

- (3 + 4) : - 3 - 4

- (3 + 4)

Distribute parentheses

= - (3) - (4)

Apply minus-plus rules

+ (- a) = - a

= - 3 - 4

= - 3 + 4 - 3 - 4

Simplify

- 3 + 4 - 3 - 4 : - 23

- 3 + 4 - 3 - 4

Add similar elements:

- 3 - 3 = - 23

= - 23 + 4 - 4

4 - 4 = 0

= - 23

= - 23

= \frac{- 2 3}{4}

Apply the fraction rule:

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

= - \frac{2 3}{4}

Cancel the common factor:

2

= - \frac{3}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = sin (x)

sin (x) = 2, sin (x) = - \frac{3}{2}

sin (x) = 2, sin (x) = - \frac{3}{2}

sin (x) = 2 :

No Solution

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - \frac{3}{2} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sin (x) = - \frac{3}{2}

General solutions for

sin (x) = - \frac{3}{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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Popular Examples

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4 cos (30 x) + 12 = 10.5

3sin^2(x)+5cos(x)=1

3 sin^{2} (x) + 5 cos (x) = 1

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2sin(x)+cos(x)=-2

2 sin (x) + cos (x) = - 2

sin(x)-sin(2x)=0,x<= 2pi,0

sin (x) - sin (2 x) = 0, x \leq 2 π, 0

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)+(sqrt(3)-4)sin(x)=2sqrt(3) ?
The general solution for 2sin^2(x)+(sqrt(3)-4)sin(x)=2sqrt(3) is x=(4pi)/3+2pin,x=(5pi)/3+2pin