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

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

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

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sin^2(θ)-2=sin(θ),-pi<= x<= 2pi

Solution

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

Solution

θ = \frac{3 π}{2}, θ = - \frac{π}{2}

Degrees

θ = 27 0^{\circ}, θ = - 9 0^{\circ}

Solution steps

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

Solve by substitution

sin^{2} (θ) - 2 = sin (θ)

Let:

sin (θ) = u

u^{2} - 2 = u

u^{2} - 2 = u : u = 2, u = - 1

u^{2} - 2 = u

Move

u

to the left side

u^{2} - 2 = u

Subtract

u

from both sides

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

Simplify

u^{2} - 2 - u = 0

u^{2} - 2 - u = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} - u - 2 = 0

Solve with the quadratic formula

u^{2} - u - 2 = 0

Quadratic Equation Formula:

For

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

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

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

(- 1)^{2} - 4 \cdot 1 \cdot (- 2) = 3

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

Apply rule

- (- a) = a

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

(- 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 1 \cdot 2 = 8

4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 8

= 1 + 8

Add the numbers:

1 + 8 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

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

Separate the solutions

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

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

\frac{- ( - 1 ) + 3}{2 \cdot 1}

Apply rule

- (- a) = a

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

Add the numbers:

1 + 3 = 4

= \frac{4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

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

\frac{- ( - 1 ) - 3}{2 \cdot 1}

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 3 = - 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 2, u = - 1

Substitute back

u = sin (θ)

sin (θ) = 2, sin (θ) = - 1

sin (θ) = 2, sin (θ) = - 1

sin (θ) = 2, - π \leq x \leq 2 π :

No Solution

sin (θ) = 2, - π \leq x \leq 2 π

- 1 \leq sin (x) \leq 1

No Solution

sin (θ) = - 1, - π \leq x \leq 2 π : θ = \frac{3 π}{2}, θ = - \frac{π}{2}

sin (θ) = - 1, - π \leq x \leq 2 π

General solutions for

sin (θ) = - 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}

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

Solutions for the range

- π \leq x \leq 2 π

θ = \frac{3 π}{2}, θ = - \frac{π}{2}

Combine all the solutions

θ = \frac{3 π}{2}, θ = - \frac{π}{2}

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Frequently Asked Questions (FAQ)

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