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Popular >

2cos(θ)=3cos(3θ)

Solution

2 cos (θ) = 3 cos (3 θ)

Solution

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 2.84874 \dots + 2 πn, θ = - 2.84874 \dots + 2 πn, θ = 0.29284 \dots + 2 πn, θ = 2 π - 0.29284 \dots + 2 πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 163.22134 \dots^{\circ} + 36 0^{\circ} n, θ = - 163.22134 \dots^{\circ} + 36 0^{\circ} n, θ = 16.77865 \dots^{\circ} + 36 0^{\circ} n, θ = 343.22134 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (θ) = 3 cos (3 θ)

Subtract

3 cos (3 θ)

from both sides

2 cos (θ) - 3 cos (3 θ) = 0

Rewrite using trig identities

2 cos (θ) - 3 cos (3 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)

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

Simplify

2 cos (θ) - 3 (4 cos^{3} (θ) - 3 cos (θ)) : 11 cos (θ) - 12 cos^{3} (θ)

2 cos (θ) - 3 (4 cos^{3} (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)

= 2 cos (θ) - 12 cos^{3} (θ) + 9 cos (θ)

Add similar elements:

2 cos (θ) + 9 cos (θ) = 11 cos (θ)

= 11 cos (θ) - 12 cos^{3} (θ)

= 11 cos (θ) - 12 cos^{3} (θ)

11 cos (θ) - 12 cos^{3} (θ) = 0

Solve by substitution

11 cos (θ) - 12 cos^{3} (θ) = 0

Let:

cos (θ) = u

11 u - 12 u^{3} = 0

11 u - 12 u^{3} = 0 : u = 0, u = - \frac{33}{6}, u = \frac{33}{6}

11 u - 12 u^{3} = 0

Factor

11 u - 12 u^{3} : - u (23 u + 11) (23 u - 11)

11 u - 12 u^{3}

Factor out common term

- u : - u (12 u^{2} - 11)

- 12 u^{3} + 11 u

Apply exponent rule:

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

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

= - 12 u^{2} u + 11 u

Factor out common term

- u

= - u (12 u^{2} - 11)

= - u (12 u^{2} - 11)

Factor

12 u^{2} - 11 : (12 u + 11) (12 u - 11)

12 u^{2} - 11

Rewrite

12 u^{2} - 11

(12 u)^{2} - (11)^{2}

12 u^{2} - 11

Apply radical rule:

a = (a)^{2}

12 = (12)^{2}

= (12)^{2} u^{2} - 11

Apply radical rule:

a = (a)^{2}

11 = (11)^{2}

= (12)^{2} u^{2} - (11)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

= (12 u)^{2} - (11)^{2}

= (12 u)^{2} - (11)^{2}

Apply Difference of Two Squares Formula:

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

(12 u)^{2} - (11)^{2} = (12 u + 11) (12 u - 11)

= (12 u + 11) (12 u - 11)

= - u (12 u + 11) (12 u - 11)

Refine

= - u (23 u + 11) (23 u - 11)

- u (23 u + 11) (23 u - 11) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 23 u + 11 = 0 or 23 u - 11 = 0

Solve

23 u + 11 = 0 : u = - \frac{33}{6}

23 u + 11 = 0

Move

11

to the right side

23 u + 11 = 0

Subtract

11

from both sides

23 u + 11 - 11 = 0 - 11

Simplify

23 u = - 11

23 u = - 11

Divide both sides by

23

23 u = - 11

Divide both sides by

23

\frac{2 3 u}{2 3} = \frac{- 11}{2 3}

Simplify

\frac{2 3 u}{2 3} = \frac{- 11}{2 3}

Simplify

\frac{2 3 u}{2 3} : u

\frac{2 3 u}{2 3}

Divide the numbers:

\frac{2}{2} = 1

= \frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 11}{2 3} : - \frac{33}{6}

\frac{- 11}{2 3}

Apply the fraction rule:

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

= - \frac{11}{2 3}

Rationalize

- \frac{11}{2 3} : - \frac{33}{6}

- \frac{11}{2 3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{11 3}{2 3 3}

113 = 33

113

Apply radical rule:

a b = a \cdot b

113 = 11 \cdot 3

= 11 \cdot 3

Multiply the numbers:

11 \cdot 3 = 33

= 33

233 = 6

233

Apply radical rule:

a a = a

33 = 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - \frac{33}{6}

= - \frac{33}{6}

u = - \frac{33}{6}

u = - \frac{33}{6}

u = - \frac{33}{6}

Solve

23 u - 11 = 0 : u = \frac{33}{6}

23 u - 11 = 0

Move

11

to the right side

23 u - 11 = 0

Add

11

to both sides

23 u - 11 + 11 = 0 + 11

Simplify

23 u = 11

23 u = 11

Divide both sides by

23

23 u = 11

Divide both sides by

23

\frac{2 3 u}{2 3} = \frac{11}{2 3}

Simplify

\frac{2 3 u}{2 3} = \frac{11}{2 3}

Simplify

\frac{2 3 u}{2 3} : u

\frac{2 3 u}{2 3}

Divide the numbers:

\frac{2}{2} = 1

= \frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{11}{2 3} : \frac{33}{6}

\frac{11}{2 3}

Multiply by the conjugate

\frac{3}{3}

= \frac{11 3}{2 3 3}

113 = 33

113

Apply radical rule:

a b = a \cdot b

113 = 11 \cdot 3

= 11 \cdot 3

Multiply the numbers:

11 \cdot 3 = 33

= 33

233 = 6

233

Apply radical rule:

a a = a

33 = 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{33}{6}

u = \frac{33}{6}

u = \frac{33}{6}

u = \frac{33}{6}

The solutions are

u = 0, u = - \frac{33}{6}, u = \frac{33}{6}

Substitute back

u = cos (θ)

cos (θ) = 0, cos (θ) = - \frac{33}{6}, cos (θ) = \frac{33}{6}

cos (θ) = 0, cos (θ) = - \frac{33}{6}, cos (θ) = \frac{33}{6}

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) = - \frac{33}{6} : θ = arccos (- \frac{33}{6}) + 2 πn, θ = - arccos (- \frac{33}{6}) + 2 πn

cos (θ) = - \frac{33}{6}

Apply trig inverse properties

cos (θ) = - \frac{33}{6}

General solutions for

cos (θ) = - \frac{33}{6}

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

θ = arccos (- \frac{33}{6}) + 2 πn, θ = - arccos (- \frac{33}{6}) + 2 πn

θ = arccos (- \frac{33}{6}) + 2 πn, θ = - arccos (- \frac{33}{6}) + 2 πn

cos (θ) = \frac{33}{6} : θ = arccos (\frac{33}{6}) + 2 πn, θ = 2 π - arccos (\frac{33}{6}) + 2 πn

cos (θ) = \frac{33}{6}

Apply trig inverse properties

cos (θ) = \frac{33}{6}

General solutions for

cos (θ) = \frac{33}{6}

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

θ = arccos (\frac{33}{6}) + 2 πn, θ = 2 π - arccos (\frac{33}{6}) + 2 πn

θ = arccos (\frac{33}{6}) + 2 πn, θ = 2 π - arccos (\frac{33}{6}) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = arccos (- \frac{33}{6}) + 2 πn, θ = - arccos (- \frac{33}{6}) + 2 πn, θ = arccos (\frac{33}{6}) + 2 πn, θ = 2 π - arccos (\frac{33}{6}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 2.84874 \dots + 2 πn, θ = - 2.84874 \dots + 2 πn, θ = 0.29284 \dots + 2 πn, θ = 2 π - 0.29284 \dots + 2 πn

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