What is the general solution for cos^2(2x)+cos(2x)-3/4 =0 ?

The general solution for cos^2(2x)+cos(2x)-3/4 =0 is x= pi/6+pin,x=(5pi)/6+pin

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cos^2(2x)+cos(2x)-3/4 =0

Solution

cos^{2} (2 x) + cos (2 x) - \frac{3}{4} = 0

Solution

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

cos^{2} (2 x) + cos (2 x) - \frac{3}{4} = 0

Solve by substitution

cos^{2} (2 x) + cos (2 x) - \frac{3}{4} = 0

Let:

cos (2 x) = u

u^{2} + u - \frac{3}{4} = 0

u^{2} + u - \frac{3}{4} = 0 : u = \frac{1}{2}, u = - \frac{3}{2}

u^{2} + u - \frac{3}{4} = 0

Multiply both sides by

4

u^{2} + u - \frac{3}{4} = 0

Multiply both sides by

4

u^{2} \cdot 4 + u \cdot 4 - \frac{3}{4} \cdot 4 = 0 \cdot 4

Simplify

4 u^{2} + 4 u - 3 = 0

4 u^{2} + 4 u - 3 = 0

Solve with the quadratic formula

4 u^{2} + 4 u - 3 = 0

Quadratic Equation Formula:

For

a = 4, b = 4, c = - 3

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 4 ( - 3 )}{2 \cdot 4}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 4 ( - 3 )}{2 \cdot 4}

4^{2} - 4 \cdot 4 (- 3) = 8

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

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

Add the numbers:

16 + 48 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- 4 \pm 8}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 4 + 8}{2 \cdot 4}, u_{2} = \frac{- 4 - 8}{2 \cdot 4}

u = \frac{- 4 + 8}{2 \cdot 4} : \frac{1}{2}

\frac{- 4 + 8}{2 \cdot 4}

Add/Subtract the numbers:

- 4 + 8 = 4

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{- 4 - 8}{2 \cdot 4} : - \frac{3}{2}

\frac{- 4 - 8}{2 \cdot 4}

Subtract the numbers:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 12}{8}

Apply the fraction rule:

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

= - \frac{12}{8}

Cancel the common factor:

4

= - \frac{3}{2}

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = - \frac{3}{2}

Substitute back

u = cos (2 x)

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

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

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

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{π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

2 x = \frac{π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

Solve

2 x = \frac{π}{3} + 2 πn : x = \frac{π}{6} + πn

2 x = \frac{π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{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{π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Solve

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{6} + πn

2 x = \frac{5 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{3} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{5 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{5 π}{3}}{2} + \frac{2 πn}{2} : \frac{5 π}{6} + πn

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{3}}{2} = \frac{5 π}{6}

\frac{\frac{5 π}{3}}{2}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{5 π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

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

No Solution

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

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(2x)+cos(2x)-3/4 =0 ?
The general solution for cos^2(2x)+cos(2x)-3/4 =0 is x= pi/6+pin,x=(5pi)/6+pin