What is the general solution for sin^2(θ)-11sin(θ)=0 ?

The general solution for sin^2(θ)-11sin(θ)=0 is θ=2pin,θ=pi+2pin

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sin^2(θ)-11sin(θ)=0

Solution

sin^{2} (θ) - 11 sin (θ) = 0

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (θ) - 11 sin (θ) = 0

Solve by substitution

sin^{2} (θ) - 11 sin (θ) = 0

Let:

sin (θ) = u

u^{2} - 11 u = 0

u^{2} - 11 u = 0 : u = 11, u = 0

u^{2} - 11 u = 0

Solve with the quadratic formula

u^{2} - 11 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 11, c = 0

u_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 11)^{2} - 4 \cdot 1 \cdot 0 = 11

(- 11)^{2} - 4 \cdot 1 \cdot 0

Apply exponent rule:

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

n

is even

(- 11)^{2} = 1 1^{2}

= 1 1^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 1 1^{2} - 0

1 1^{2} - 0 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

assuming

a \geq 0

= 11

u_{1, 2} = \frac{- ( - 11 ) \pm 11}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 11 ) + 11}{2 \cdot 1}, u_{2} = \frac{- ( - 11 ) - 11}{2 \cdot 1}

u = \frac{- ( - 11 ) + 11}{2 \cdot 1} : 11

\frac{- ( - 11 ) + 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{11 + 11}{2 \cdot 1}

Add the numbers:

11 + 11 = 22

= \frac{22}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{22}{2}

Divide the numbers:

\frac{22}{2} = 11

= 11

u = \frac{- ( - 11 ) - 11}{2 \cdot 1} : 0

\frac{- ( - 11 ) - 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{11 - 11}{2 \cdot 1}

Subtract the numbers:

11 - 11 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

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

= 0

The solutions to the quadratic equation are:

u = 11, u = 0

Substitute back

u = sin (θ)

sin (θ) = 11, sin (θ) = 0

sin (θ) = 11, sin (θ) = 0

sin (θ) = 11 :

No Solution

sin (θ) = 11

- 1 \leq sin (x) \leq 1

No Solution

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

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

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(θ)-11sin(θ)=0 ?
The general solution for sin^2(θ)-11sin(θ)=0 is θ=2pin,θ=pi+2pin