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

The general solution for 2sin(x)=sqrt(cos(2x)+2) is x= pi/4+2pin,x=(3pi)/4+2pin

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2sin(x)=sqrt(cos(2x)+2)

Solution

2 sin (x) = cos (2 x) + 2

Solution

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

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) = cos (2 x) + 2

Subtract

cos (2 x) + 2

from both sides

2 sin (x) - cos (2 x) + 2 = 0

Rewrite using trig identities

- 2 + cos (2 x) + 2 sin (x)

Use the Double Angle identity:

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

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

Simplify

= - - 2 sin^{2} (x) + 3 + 2 sin (x)

- 3 - 2 sin^{2} (x) + 2 sin (x) = 0

Solve by substitution

- 3 - 2 sin^{2} (x) + 2 sin (x) = 0

Let:

sin (x) = u

- 3 - 2 u^{2} + 2 u = 0

- 3 - 2 u^{2} + 2 u = 0 : u = \frac{1}{2}

- 3 - 2 u^{2} + 2 u = 0

Remove square roots

- 3 - 2 u^{2} + 2 u = 0

Subtract

2 u

from both sides

- 3 - 2 u^{2} + 2 u - 2 u = 0 - 2 u

Simplify

- 3 - 2 u^{2} = - 2 u

Square both sides:

3 - 2 u^{2} = 4 u^{2}

- 3 - 2 u^{2} + 2 u = 0

(- 3 - 2 u^{2})^{2} = (- 2 u)^{2}

Expand

(- 3 - 2 u^{2})^{2} : 3 - 2 u^{2}

(- 3 - 2 u^{2})^{2}

Apply exponent rule:

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

n

is even

(- 3 - 2 u^{2})^{2} = (3 - 2 u^{2})^{2}

= (3 - 2 u^{2})^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= (3 - 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

= 3 - 2 u^{2}

Expand

(- 2 u)^{2} : 4 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} = 4

= 4 u^{2}

3 - 2 u^{2} = 4 u^{2}

3 - 2 u^{2} = 4 u^{2}

3 - 2 u^{2} = 4 u^{2}

Solve

3 - 2 u^{2} = 4 u^{2} : u = \frac{1}{2}, u = - \frac{1}{2}

3 - 2 u^{2} = 4 u^{2}

Move

3

to the right side

3 - 2 u^{2} = 4 u^{2}

Subtract

3

from both sides

3 - 2 u^{2} - 3 = 4 u^{2} - 3

Simplify

- 2 u^{2} = 4 u^{2} - 3

- 2 u^{2} = 4 u^{2} - 3

Move

4 u^{2}

to the left side

- 2 u^{2} = 4 u^{2} - 3

Subtract

4 u^{2}

from both sides

- 2 u^{2} - 4 u^{2} = 4 u^{2} - 3 - 4 u^{2}

Simplify

- 6 u^{2} = - 3

- 6 u^{2} = - 3

Divide both sides by

- 6

- 6 u^{2} = - 3

Divide both sides by

- 6

\frac{- 6 u ^{2}}{- 6} = \frac{- 3}{- 6}

Simplify

\frac{- 6 u ^{2}}{- 6} = \frac{- 3}{- 6}

Simplify

\frac{- 6 u ^{2}}{- 6} : u^{2}

\frac{- 6 u ^{2}}{- 6}

Apply the fraction rule:

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

= \frac{6 u ^{2}}{6}

Divide the numbers:

\frac{6}{6} = 1

= u^{2}

Simplify

\frac{- 3}{- 6} : \frac{1}{2}

\frac{- 3}{- 6}

Apply the fraction rule:

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

= \frac{3}{6}

Cancel the common factor:

3

= \frac{1}{2}

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

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

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

For

x^{2} = f (a)

the solutions are

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

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

- 3 - 2 u^{2} + 2 u = 0

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

Plug in

u = \frac{1}{2} :

True

- 3 - 2 (\frac{1}{2})^{2} + 2 \frac{1}{2} = 0

- 3 - 2 (\frac{1}{2})^{2} + 2 \frac{1}{2} = 0

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

3 - 2 (\frac{1}{2})^{2} = 2

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

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

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{1}{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

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

2 \frac{1}{2} = 2

2 \frac{1}{2}

Apply exponent rule:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

2

= 2

= 2

= - 2 + 2

Add similar elements:

- 2 + 2 = 0

= 0

0 = 0

True

Plug in

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

False

- 3 - 2 (- \frac{1}{2})^{2} + 2 (- \frac{1}{2}) = 0

- 3 - 2 (- \frac{1}{2})^{2} + 2 (- \frac{1}{2}) = - 22

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

Remove parentheses:

(- a) = - a

= - 3 - 2 (- \frac{1}{2})^{2} - 2 \frac{1}{2}

3 - 2 (- \frac{1}{2})^{2} = 2

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

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

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

(- \frac{1}{2})^{2} = \frac{1}{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 radical rule:

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

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

Apply exponent rule:

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

= (\frac{1}{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

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

2 \frac{1}{2} = 2

2 \frac{1}{2}

Apply exponent rule:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

2

= 2

= 2

= - 2 - 2

Add similar elements:

- 2 - 2 = - 22

= - 22

- 22 = 0

False

The solution is

u = \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \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}

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

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

Combine all the solutions

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

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tan (x) = \frac{7}{12}

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cos (x) = 1 - sin (x)

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tan (x) = 7

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)=sqrt(cos(2x)+2) ?
The general solution for 2sin(x)=sqrt(cos(2x)+2) is x= pi/4+2pin,x=(3pi)/4+2pin