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

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

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

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

3

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

= - 3 - (cos (2 x) - 1) - 3 cos (2 x)

Simplify

- 3 - (cos (2 x) - 1) - 3 cos (2 x) : - 4 cos (2 x) - 2

- 3 - (cos (2 x) - 1) - 3 cos (2 x)

- (cos (2 x) - 1) : - cos (2 x) + 1

- (cos (2 x) - 1)

Distribute parentheses

= - (cos (2 x)) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - cos (2 x) + 1

= - 3 - cos (2 x) + 1 - 3 cos (2 x)

Simplify

- 3 - cos (2 x) + 1 - 3 cos (2 x) : - 4 cos (2 x) - 2

- 3 - cos (2 x) + 1 - 3 cos (2 x)

Group like terms

= - cos (2 x) - 3 cos (2 x) - 3 + 1

Add similar elements:

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

= - 4 cos (2 x) - 3 + 1

Add/Subtract the numbers:

- 3 + 1 = - 2

= - 4 cos (2 x) - 2

= - 4 cos (2 x) - 2

= - 4 cos (2 x) - 2

- 2 - 4 cos (2 x) = 0

Move

2

to the right side

- 2 - 4 cos (2 x) = 0

Add

2

to both sides

- 2 - 4 cos (2 x) + 2 = 0 + 2

Simplify

- 4 cos (2 x) = 2

- 4 cos (2 x) = 2

Divide both sides by

- 4

- 4 cos (2 x) = 2

Divide both sides by

- 4

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

Simplify

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

Simplify

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

\frac{- 4 cos ( 2 x )}{- 4}

Apply the fraction rule:

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

= \frac{4 cos ( 2 x )}{4}

Divide the numbers:

\frac{4}{4} = 1

= cos (2 x)

Simplify

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

\frac{2}{- 4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

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

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

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

General solutions for

cos (2 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}

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

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

Solve

2 x = \frac{2 π}{3} + 2 πn : x = \frac{π}{3} + πn

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{3} + πn

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

Solve

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

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2} : \frac{2 π}{3} + πn

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{4 π}{3}}{2} = \frac{2 π}{3}

\frac{\frac{4 π}{3}}{2}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

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

Graph

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Frequently Asked Questions (FAQ)

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