What is the general solution for 3cos^2(x)-cos(2x)=1 ?

The general solution for 3cos^2(x)-cos(2x)=1 is x= pi/2+2pin,x=(3pi)/2+2pin

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

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

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

3 cos^{2} (x) - cos (2 x) - 1 = 0

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

- 1 - (2 cos^{2} (x) - 1) + 3 cos^{2} (x) : cos^{2} (x)

- 1 - (2 cos^{2} (x) - 1) + 3 cos^{2} (x)

- (2 cos^{2} (x) - 1) : - 2 cos^{2} (x) + 1

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Simplify

- 1 - 2 cos^{2} (x) + 1 + 3 cos^{2} (x) : cos^{2} (x)

- 1 - 2 cos^{2} (x) + 1 + 3 cos^{2} (x)

Group like terms

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

Add similar elements:

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

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

- 1 + 1 = 0

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

cos^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

cos (x) = 0

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

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

sin (2 x) + sin (4 x) = cos (x)

25 cos^{2} (x) - 10 cos (x) - 1 = 0

1 = 2 cos (2 θ)

csc (θ) - sin (θ) = cot (θ) csc (θ)

2 cos^{2} (θ) + 2 cos (θ) = 0

What is the general solution for 3cos^2(x)-cos(2x)=1 ?
The general solution for 3cos^2(x)-cos(2x)=1 is x= pi/2+2pin,x=(3pi)/2+2pin