What is the general solution for sqrt(2)sin^2(θ)-sin(θ)=0,0<= θ<2pi ?

Q: What is the general solution for sqrt(2)sin^2(θ)-sin(θ)=0,0<= θ<2pi ?

The general solution for sqrt(2)sin^2(θ)-sin(θ)=0,0<= θ<2pi is θ= pi/4 ,θ=(3pi)/4 ,θ=0,θ=pi

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sqrt(2)sin^2(θ)-sin(θ)=0,0<= θ<2pi

Solution

2 sin^{2} (θ) - sin (θ) = 0, 0 \leq θ < 2 π

Solution

θ = \frac{π}{4}, θ = \frac{3 π}{4}, θ = 0, θ = π

Degrees

θ = 4 5^{\circ}, θ = 13 5^{\circ}, θ = 0^{\circ}, θ = 18 0^{\circ}

Solution steps

2 sin^{2} (θ) - sin (θ) = 0, 0 \leq θ < 2 π

Solve by substitution

2 sin^{2} (θ) - sin (θ) = 0

Let:

sin (θ) = u

2 u^{2} - u = 0

2 u^{2} - u = 0 : u = \frac{2}{2}, u = 0

2 u^{2} - u = 0

Solve with the quadratic formula

2 u^{2} - u = 0

Quadratic Equation Formula:

For

a = 2, b = - 1, c = 0

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

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

(- 1)^{2} - 42 \cdot 0 = 1

(- 1)^{2} - 42 \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

42 \cdot 0 = 0

42 \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 2}

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

1 + 1 = 2

= \frac{2}{2 2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{2}

Rationalize

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2 2}

Apply rule

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

= 0

The solutions to the quadratic equation are:

u = \frac{2}{2}, u = 0

Substitute back

u = sin (θ)

sin (θ) = \frac{2}{2}, sin (θ) = 0

sin (θ) = \frac{2}{2}, sin (θ) = 0

sin (θ) = \frac{2}{2}, 0 \leq θ < 2 π : θ = \frac{π}{4}, θ = \frac{3 π}{4}

sin (θ) = \frac{2}{2}, 0 \leq θ < 2 π

General solutions for

sin (θ) = \frac{2}{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}

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

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

Solutions for the range

0 \leq θ < 2 π

θ = \frac{π}{4}, θ = \frac{3 π}{4}

sin (θ) = 0, 0 \leq θ < 2 π : θ = 0, θ = π

sin (θ) = 0, 0 \leq θ < 2 π

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Solutions for the range

0 \leq θ < 2 π

θ = 0, θ = π

Combine all the solutions

θ = \frac{π}{4}, θ = \frac{3 π}{4}, θ = 0, θ = π

Popular Examples

cos(x^2)= 1/2

cos (x^{2}) = \frac{1}{2}

sec^2(x)-7sec(x)=0

sec^{2} (x) - 7 sec (x) = 0

e^{pi/2}-e^{arctan(1+x^2)}=0

e^{\frac{π}{2}} - e^{a r c t a n (1 + x^{2})} = 0

3cot(θ)-7=0

3 cot (θ) - 7 = 0

cos(x)= 1249/1250

cos (x) = \frac{1249}{1250}

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sin^2(θ)-sin(θ)=0,0<= θ<2pi ?
The general solution for sqrt(2)sin^2(θ)-sin(θ)=0,0<= θ<2pi is θ= pi/4 ,θ=(3pi)/4 ,θ=0,θ=pi