What is the general solution for 6sin^2(x)-9sin(x)+3=0 ?

The general solution for 6sin^2(x)-9sin(x)+3=0 is x= pi/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

6sin^2(x)-9sin(x)+3=0

Solution

6 sin^{2} (x) - 9 sin (x) + 3 = 0

Solution

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

6 sin^{2} (x) - 9 sin (x) + 3 = 0

Solve by substitution

6 sin^{2} (x) - 9 sin (x) + 3 = 0

Let:

sin (x) = u

6 u^{2} - 9 u + 3 = 0

6 u^{2} - 9 u + 3 = 0 : u = 1, u = \frac{1}{2}

6 u^{2} - 9 u + 3 = 0

Solve with the quadratic formula

6 u^{2} - 9 u + 3 = 0

Quadratic Equation Formula:

For

a = 6, b = - 9, c = 3

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

(- 9)^{2} - 4 \cdot 6 \cdot 3 = 3

(- 9)^{2} - 4 \cdot 6 \cdot 3

Apply exponent rule:

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

n

is even

(- 9)^{2} = 9^{2}

= 9^{2} - 4 \cdot 6 \cdot 3

Multiply the numbers:

4 \cdot 6 \cdot 3 = 72

= 9^{2} - 72

9^{2} = 81

= 81 - 72

Subtract the numbers:

81 - 72 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 9 ) \pm 3}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 9 ) + 3}{2 \cdot 6}, u_{2} = \frac{- ( - 9 ) - 3}{2 \cdot 6}

u = \frac{- ( - 9 ) + 3}{2 \cdot 6} : 1

\frac{- ( - 9 ) + 3}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{9 + 3}{2 \cdot 6}

Add the numbers:

9 + 3 = 12

= \frac{12}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{12}{12}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 9 ) - 3}{2 \cdot 6} : \frac{1}{2}

\frac{- ( - 9 ) - 3}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{9 - 3}{2 \cdot 6}

Subtract the numbers:

9 - 3 = 6

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

Multiply the numbers:

2 \cdot 6 = 12

= \frac{6}{12}

Cancel the common factor:

6

= \frac{1}{2}

The solutions to the quadratic equation are:

u = 1, u = \frac{1}{2}

Substitute back

u = sin (x)

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

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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

Combine all the solutions

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

Popular Examples

cos(2x)=0.25

cos (2 x) = 0.25

3cos^2(x)=2

3 cos^{2} (x) = 2

sqrt(3)tan(θ)=2sin(θ)

3 tan (θ) = 2 sin (θ)

sin(x)=-0.1

sin (x) = - 0.1

189.4=100tan^2(45+θ/2)

189.4 = 100 tan^{2} (4 5^{\circ} + \frac{θ}{2})

Frequently Asked Questions (FAQ)

What is the general solution for 6sin^2(x)-9sin(x)+3=0 ?
The general solution for 6sin^2(x)-9sin(x)+3=0 is x= pi/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin