What is the general solution for cos(3θ)=4cos(3θ)-3cos(θ) ?

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

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cos(3θ)=4cos(3θ)-3cos(θ)

Solution

cos (3 θ) = 4 cos (3 θ) - 3 cos (θ)

Solution

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

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (3 θ) = 4 cos (3 θ) - 3 cos (θ)

Subtract

4 cos (3 θ) - 3 cos (θ)

from both sides

- 3 cos (3 θ) + 3 cos (θ) = 0

Rewrite using trig identities

- 3 cos (3 θ) + 3 cos (θ)

cos (3 θ) = 4 cos^{3} (θ) - 3 cos (θ)

cos (3 θ)

Rewrite using trig identities

cos (3 θ)

Rewrite as

= cos (2 θ + θ)

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 θ) cos (θ) - sin (2 θ) sin (θ)

Use the Double Angle identity:

sin (2 θ) = 2 sin (θ) cos (θ)

= cos (2 θ) cos (θ) - 2 sin (θ) cos (θ) sin (θ)

Simplify

cos (2 θ) cos (θ) - 2 sin (θ) cos (θ) sin (θ) : cos (θ) cos (2 θ) - 2 sin^{2} (θ) cos (θ)

cos (2 θ) cos (θ) - 2 sin (θ) cos (θ) sin (θ)

2 sin (θ) cos (θ) sin (θ) = 2 sin^{2} (θ) cos (θ)

2 sin (θ) cos (θ) sin (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= 2 cos (θ) sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= 2 cos (θ) sin^{2} (θ)

= cos (θ) cos (2 θ) - 2 sin^{2} (θ) cos (θ)

= cos (θ) cos (2 θ) - 2 sin^{2} (θ) cos (θ)

= cos (θ) cos (2 θ) - 2 sin^{2} (θ) cos (θ)

Use the Double Angle identity:

cos (2 θ) = 2 cos^{2} (θ) - 1

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

Use the Pythagorean identity:

cos^{2} (θ) + sin^{2} (θ) = 1

sin^{2} (θ) = 1 - cos^{2} (θ)

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

Expand

(2 cos^{2} (θ) - 1) cos (θ) - 2 (1 - cos^{2} (θ)) cos (θ) : 4 cos^{3} (θ) - 3 cos (θ)

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

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

Expand

cos (θ) (2 cos^{2} (θ) - 1) : 2 cos^{3} (θ) - cos (θ)

cos (θ) (2 cos^{2} (θ) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = cos (θ), b = 2 cos^{2} (θ), c = 1

= cos (θ) 2 cos^{2} (θ) - cos (θ) 1

= 2 cos^{2} (θ) cos (θ) - 1 cos (θ)

Simplify

2 cos^{2} (θ) cos (θ) - 1 \cdot cos (θ) : 2 cos^{3} (θ) - cos (θ)

2 cos^{2} (θ) cos (θ) - 1 cos (θ)

2 cos^{2} (θ) cos (θ) = 2 cos^{3} (θ)

2 cos^{2} (θ) cos (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (θ) cos (θ) = cos^{2 + 1} (θ)

= 2 cos^{2 + 1} (θ)

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (θ)

1 \cdot cos (θ) = cos (θ)

1 cos (θ)

Multiply:

1 \cdot cos (θ) = cos (θ)

= cos (θ)

= 2 cos^{3} (θ) - cos (θ)

= 2 cos^{3} (θ) - cos (θ)

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

Expand

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

- 2 cos (θ) (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 2 cos (θ), b = 1, c = cos^{2} (θ)

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

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 cos (θ) + 2 cos^{2} (θ) cos (θ)

Simplify

- 2 \cdot 1 \cdot cos (θ) + 2 cos^{2} (θ) cos (θ) : - 2 cos (θ) + 2 cos^{3} (θ)

- 2 \cdot 1 cos (θ) + 2 cos^{2} (θ) cos (θ)

2 \cdot 1 \cdot cos (θ) = 2 cos (θ)

2 \cdot 1 cos (θ)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (θ)

2 cos^{2} (θ) cos (θ) = 2 cos^{3} (θ)

2 cos^{2} (θ) cos (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (θ) cos (θ) = cos^{2 + 1} (θ)

= 2 cos^{2 + 1} (θ)

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (θ)

= - 2 cos (θ) + 2 cos^{3} (θ)

= - 2 cos (θ) + 2 cos^{3} (θ)

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

Simplify

2 cos^{3} (θ) - cos (θ) - 2 cos (θ) + 2 cos^{3} (θ) : 4 cos^{3} (θ) - 3 cos (θ)

2 cos^{3} (θ) - cos (θ) - 2 cos (θ) + 2 cos^{3} (θ)

Group like terms

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

Add similar elements:

2 cos^{3} (θ) + 2 cos^{3} (θ) = 4 cos^{3} (θ)

= 4 cos^{3} (θ) - cos (θ) - 2 cos (θ)

Add similar elements:

- cos (θ) - 2 cos (θ) = - 3 cos (θ)

= 4 cos^{3} (θ) - 3 cos (θ)

= 4 cos^{3} (θ) - 3 cos (θ)

= 4 cos^{3} (θ) - 3 cos (θ)

= - 3 (4 cos^{3} (θ) - 3 cos (θ)) + 3 cos (θ)

Simplify

- 3 (4 cos^{3} (θ) - 3 cos (θ)) + 3 cos (θ) : - 12 cos^{3} (θ) + 12 cos (θ)

- 3 (4 cos^{3} (θ) - 3 cos (θ)) + 3 cos (θ)

Expand

- 3 (4 cos^{3} (θ) - 3 cos (θ)) : - 12 cos^{3} (θ) + 9 cos (θ)

- 3 (4 cos^{3} (θ) - 3 cos (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 4 cos^{3} (θ), c = 3 cos (θ)

= - 3 \cdot 4 cos^{3} (θ) - (- 3) \cdot 3 cos (θ)

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 4 cos^{3} (θ) + 3 \cdot 3 cos (θ)

Simplify

- 3 \cdot 4 cos^{3} (θ) + 3 \cdot 3 cos (θ) : - 12 cos^{3} (θ) + 9 cos (θ)

- 3 \cdot 4 cos^{3} (θ) + 3 \cdot 3 cos (θ)

Multiply the numbers:

3 \cdot 4 = 12

= - 12 cos^{3} (θ) + 3 \cdot 3 cos (θ)

Multiply the numbers:

3 \cdot 3 = 9

= - 12 cos^{3} (θ) + 9 cos (θ)

= - 12 cos^{3} (θ) + 9 cos (θ)

= - 12 cos^{3} (θ) + 9 cos (θ) + 3 cos (θ)

Add similar elements:

9 cos (θ) + 3 cos (θ) = 12 cos (θ)

= - 12 cos^{3} (θ) + 12 cos (θ)

= - 12 cos^{3} (θ) + 12 cos (θ)

12 cos (θ) - 12 cos^{3} (θ) = 0

Solve by substitution

12 cos (θ) - 12 cos^{3} (θ) = 0

Let:

cos (θ) = u

12 u - 12 u^{3} = 0

12 u - 12 u^{3} = 0 : u = 0, u = - 1, u = 1

12 u - 12 u^{3} = 0

Factor

12 u - 12 u^{3} : - 12 u (u + 1) (u - 1)

12 u - 12 u^{3}

Factor out common term

- 12 u : - 12 u (u^{2} - 1)

- 12 u^{3} + 12 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= - 12 u^{2} u + 12 u

Factor out common term

- 12 u

= - 12 u (u^{2} - 1)

= - 12 u (u^{2} - 1)

Factor

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

Rewrite

1

1^{2}

= u^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= - 12 u (u + 1) (u - 1)

- 12 u (u + 1) (u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u + 1 = 0 or u - 1 = 0

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

The solutions are

u = 0, u = - 1, u = 1

Substitute back

u = cos (θ)

cos (θ) = 0, cos (θ) = - 1, cos (θ) = 1

cos (θ) = 0, cos (θ) = - 1, cos (θ) = 1

cos (θ) = 0 : θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

cos (θ) = - 1 : θ = π + 2 πn

cos (θ) = - 1

General solutions for

cos (θ) = - 1

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}

θ = π + 2 πn

θ = π + 2 πn

cos (θ) = 1 : θ = 2 πn

cos (θ) = 1

General solutions for

cos (θ) = 1

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}

θ = 0 + 2 πn

θ = 0 + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

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