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

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

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cos^2(3x)=1

Solution

cos^{2} (3 x) = 1

Solution

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

Degrees

x = 0^{\circ} + 12 0^{\circ} n, x = 6 0^{\circ} + 12 0^{\circ} n

Solution steps

cos^{2} (3 x) = 1

Solve by substitution

cos^{2} (3 x) = 1

Let:

cos (3 x) = u

u^{2} = 1

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

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Substitute back

u = cos (3 x)

cos (3 x) = 1, cos (3 x) = - 1

cos (3 x) = 1, cos (3 x) = - 1

cos (3 x) = 1 : x = \frac{2 πn}{3}

cos (3 x) = 1

General solutions for

cos (3 x) = 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}

3 x = 0 + 2 πn

3 x = 0 + 2 πn

Solve

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

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

cos (3 x) = - 1

General solutions for

cos (3 x) = - 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}

3 x = π + 2 πn

3 x = π + 2 πn

Solve

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

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

Combine all the solutions

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

Popular Examples

tan(pi/4-x)+cot(pi/4-x)=4

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

7sin(θ)-1=5sin(θ)

7 sin (θ) - 1 = 5 sin (θ)

csc(θ)=-7/3

csc (θ) = - \frac{7}{3}

sin(θ)= 5/8

sin (θ) = \frac{5}{8}

arctan(2x)+arctan(x)= pi/4

arctan (2 x) + arctan (x) = \frac{π}{4}

Frequently Asked Questions (FAQ)

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