What is the general solution for 8sin^2(x)=sin(x) ?

The general solution for 8sin^2(x)=sin(x) is x=0.12532…+2pin,x=pi-0.12532…+2pin,x=2pin,x=pi+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

8sin^2(x)=sin(x)

Solution

8 sin^{2} (x) = sin (x)

Solution

x = 0.12532 \dots + 2 πn, x = π - 0.12532 \dots + 2 πn, x = 2 πn, x = π + 2 πn

Degrees

x = 7.18075 \dots^{\circ} + 36 0^{\circ} n, x = 172.81924 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

8 sin^{2} (x) = sin (x)

Solve by substitution

8 sin^{2} (x) = sin (x)

Let:

sin (x) = u

8 u^{2} = u

8 u^{2} = u : u = \frac{1}{8}, u = 0

8 u^{2} = u

Move

u

to the left side

8 u^{2} = u

Subtract

u

from both sides

8 u^{2} - u = u - u

Simplify

8 u^{2} - u = 0

8 u^{2} - u = 0

Solve with the quadratic formula

8 u^{2} - u = 0

Quadratic Equation Formula:

For

a = 8, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 8 \cdot 0}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 8 \cdot 0}{2 \cdot 8}

(- 1)^{2} - 4 \cdot 8 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 8 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 8 \cdot 0 = 0

4 \cdot 8 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- ( - 1 ) \pm 1}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 1}{2 \cdot 8}, u_{2} = \frac{- ( - 1 ) - 1}{2 \cdot 8}

u = \frac{- ( - 1 ) + 1}{2 \cdot 8} : \frac{1}{8}

\frac{- ( - 1 ) + 1}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{1 + 1}{2 \cdot 8}

Add the numbers:

1 + 1 = 2

= \frac{2}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{2}{16}

Cancel the common factor:

2

= \frac{1}{8}

u = \frac{- ( - 1 ) - 1}{2 \cdot 8} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{1 - 1}{2 \cdot 8}

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{0}{16}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{1}{8}, u = 0

Substitute back

u = sin (x)

sin (x) = \frac{1}{8}, sin (x) = 0

sin (x) = \frac{1}{8}, sin (x) = 0

sin (x) = \frac{1}{8} : x = arcsin (\frac{1}{8}) + 2 πn, x = π - arcsin (\frac{1}{8}) + 2 πn

sin (x) = \frac{1}{8}

Apply trig inverse properties

sin (x) = \frac{1}{8}

General solutions for

sin (x) = \frac{1}{8}

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

x = arcsin (\frac{1}{8}) + 2 πn, x = π - arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn, x = π - arcsin (\frac{1}{8}) + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combine all the solutions

x = arcsin (\frac{1}{8}) + 2 πn, x = π - arcsin (\frac{1}{8}) + 2 πn, x = 2 πn, x = π + 2 πn

Show solutions in decimal form

x = 0.12532 \dots + 2 πn, x = π - 0.12532 \dots + 2 πn, x = 2 πn, x = π + 2 πn

Popular Examples

6cos(x)+6sin(x)=3sqrt(6)

6 cos (x) + 6 sin (x) = 36

3sin(x)+1=0

3 sin (x) + 1 = 0

(tan(x)+1)(cos(x)-1)=0

(tan (x) + 1) (cos (x) - 1) = 0

cot(θ)=-1/(sqrt(3))

cot (θ) = - \frac{1}{3}

3sqrt(2)cos(θ)+3=6

32 cos (θ) + 3 = 6

Frequently Asked Questions (FAQ)

What is the general solution for 8sin^2(x)=sin(x) ?
The general solution for 8sin^2(x)=sin(x) is x=0.12532…+2pin,x=pi-0.12532…+2pin,x=2pin,x=pi+2pin