What is the general solution for sqrt(3-2cos(x))=2 ?

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

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Popular >

sqrt(3-2cos(x))=2

Solution

3 - 2 cos (x) = 2

Solution

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

Solution steps

3 - 2 cos (x) = 2

Square both sides:

3 - 2 cos (x) = 4

3 - 2 cos (x) = 2

(3 - 2 cos (x))^{2} = 2^{2}

Expand

(3 - 2 cos (x))^{2} : 3 - 2 cos (x)

(3 - 2 cos (x))^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= (3 - 2 cos (x))^{\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 cos (x)

Expand

2^{2} : 4

2^{2}

2^{2} = 4

= 4

3 - 2 cos (x) = 4

3 - 2 cos (x) = 4

Solve

3 - 2 cos (x) = 4 : cos (x) = - \frac{1}{2}

3 - 2 cos (x) = 4

Move

3

to the right side

3 - 2 cos (x) = 4

Subtract

3

from both sides

3 - 2 cos (x) - 3 = 4 - 3

Simplify

- 2 cos (x) = 1

- 2 cos (x) = 1

Divide both sides by

- 2

- 2 cos (x) = 1

Divide both sides by

- 2

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

Simplify

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

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

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

Verify Solutions:

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

True

Check the solutions by plugging them into

3 - 2 cos (x) = 2

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

Plug in

cos (x) = - \frac{1}{2} :

True

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

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

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

Apply rule

- (- a) = a

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

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

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

Add the numbers:

3 + 1 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

2 = 2

True

The solution is

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Popular Examples

tan(θ)=-8/5

tan (θ) = - \frac{8}{5}

sin(θ)-0.1*cos(θ)=0.56632

sin (θ) - 0.1 \cdot cos (θ) = 0.56632

3=2cos^2(x)+3sin(x)

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

-6cos^2(θ)=0

- 6 cos^{2} (θ) = 0

arctan(x)=60

arctan (x) = 60

Frequently Asked Questions (FAQ)

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