What is the general solution for sqrt(2)sin(θ)=sqrt(cos(2θ)) ?

The general solution for sqrt(2)sin(θ)=sqrt(cos(2θ)) is θ= pi/6+2pin,θ=(5pi)/6+2pin

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sqrt(2)sin(θ)=sqrt(cos(2θ))

Solution

2 sin (θ) = cos (2 θ)

Solution

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

Degrees

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

Solution steps

2 sin (θ) = cos (2 θ)

Subtract

cos (2 θ)

from both sides

2 sin (θ) - cos (2 θ) = 0

Rewrite using trig identities

- cos (2 θ) + sin (θ) 2

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= - 1 - 2 sin^{2} (θ) + 2 sin (θ)

- 1 - 2 sin^{2} (θ) + sin (θ) 2 = 0

Solve by substitution

- 1 - 2 sin^{2} (θ) + sin (θ) 2 = 0

Let:

sin (θ) = u

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

- 1 - 2 u^{2} + u 2 = 0 : u = \frac{1}{2}

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

Remove square roots

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

Subtract

u 2

from both sides

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

Simplify

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

Square both sides:

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

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

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

Expand

(- 1 - 2 u^{2})^{2} : 1 - 2 u^{2}

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

Apply exponent rule:

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

n

is even

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

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

Apply radical rule:

a = a^{\frac{1}{2}}

= ((1 - 2 u^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 1 - 2 u^{2}

Expand

(- 2 u)^{2} : 2 u^{2}

(- 2 u)^{2}

Apply exponent rule:

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

n

is even

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

= (2 u)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= (2)^{2} u^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2

= 2 u^{2}

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

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

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

Solve

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Move

2 u^{2}

to the left side

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

Subtract

2 u^{2}

from both sides

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

Simplify

- 4 u^{2} = - 1

- 4 u^{2} = - 1

Divide both sides by

- 4

- 4 u^{2} = - 1

Divide both sides by

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- 1}{- 4}

Simplify

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{4}, u = - \frac{1}{4}

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

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

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

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

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

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

Verify Solutions:

u = \frac{1}{2}

True

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

False

Check the solutions by plugging them into

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

Remove the ones that don't agree with the equation.

Plug in

u = \frac{1}{2} :

True

- 1 - 2 (\frac{1}{2})^{2} + (\frac{1}{2}) 2 = 0

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

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

Remove parentheses:

(a) = a

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

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

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

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

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

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{2 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{2 ^{2}}

Cancel the common factor:

2

= \frac{1}{2}

= 1 - \frac{1}{2}

Join

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

1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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

\frac{1}{2} 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiply:

1 \cdot 2 = 2

= \frac{2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= \frac{1}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{1}{2}

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

- \frac{1}{2} + \frac{1}{2} = 0

True

Plug in

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

False

- 1 - 2 (- \frac{1}{2})^{2} + (- \frac{1}{2}) 2 = 0

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

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

Remove parentheses:

(- a) = - a

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

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

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

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

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

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

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{2 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{2 ^{2}}

Cancel the common factor:

2

= \frac{1}{2}

= 1 - \frac{1}{2}

Join

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

1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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

\frac{1}{2} 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiply:

1 \cdot 2 = 2

= \frac{2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= \frac{1}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{1}{2}

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

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

False

The solution is

u = \frac{1}{2}

Substitute back

u = sin (θ)

sin (θ) = \frac{1}{2}

sin (θ) = \frac{1}{2}

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = \frac{1}{2}

General solutions for

sin (θ) = \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}

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

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

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sin(θ)=sqrt(cos(2θ)) ?
The general solution for sqrt(2)sin(θ)=sqrt(cos(2θ)) is θ= pi/6+2pin,θ=(5pi)/6+2pin