What is the general solution for -cos(2x)=2cos(2x)-1 ?

The general solution for -cos(2x)=2cos(2x)-1 is x=(1.23095…)/2+pin,x=pi-(1.23095…)/2+pin

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Popular >

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

Solution

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

Solution

x = \frac{1.23095 \dots}{2} + πn, x = π - \frac{1.23095 \dots}{2} + πn

Degrees

x = 35.26438 \dots^{\circ} + 18 0^{\circ} n, x = 144.73561 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (2 x) = u

- u = 2 u - 1

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

- u = 2 u - 1

Move

2 u

to the left side

- u = 2 u - 1

Subtract

2 u

from both sides

- u - 2 u = 2 u - 1 - 2 u

Simplify

- 3 u = - 1

- 3 u = - 1

Divide both sides by

- 3

- 3 u = - 1

Divide both sides by

- 3

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

Simplify

u = \frac{1}{3}

u = \frac{1}{3}

Substitute back

u = cos (2 x)

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

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

cos (2 x) = \frac{1}{3} : x = \frac{arccos ( \frac{1}{3} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x = arccos (\frac{1}{3}) + 2 πn, 2 x = 2 π - arccos (\frac{1}{3}) + 2 πn

2 x = arccos (\frac{1}{3}) + 2 πn, 2 x = 2 π - arccos (\frac{1}{3}) + 2 πn

Solve

2 x = arccos (\frac{1}{3}) + 2 πn : x = \frac{arccos ( \frac{1}{3} )}{2} + πn

2 x = arccos (\frac{1}{3}) + 2 πn

Divide both sides by

2

2 x = arccos (\frac{1}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( \frac{1}{3} )}{2} + πn

x = \frac{arccos ( \frac{1}{3} )}{2} + πn

Solve

2 x = 2 π - arccos (\frac{1}{3}) + 2 πn : x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

2 x = 2 π - arccos (\frac{1}{3}) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (\frac{1}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

x = \frac{arccos ( \frac{1}{3} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

Combine all the solutions

x = \frac{arccos ( \frac{1}{3} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{3} )}{2} + πn

Show solutions in decimal form

x = \frac{1.23095 \dots}{2} + πn, x = π - \frac{1.23095 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for -cos(2x)=2cos(2x)-1 ?
The general solution for -cos(2x)=2cos(2x)-1 is x=(1.23095…)/2+pin,x=pi-(1.23095…)/2+pin