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Popular Trigonometry >

sin(2x)-sin(4x)+sin(6x)=0

Solution

sin (2 x) - sin (4 x) + sin (6 x) = 0

Solution

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{π + 6 πn}{6}, x = \frac{5 π + 6 πn}{6}, x = \frac{π + 4 πn}{4}, x = \frac{3 π + 4 πn}{4}

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 x) - sin (4 x) + sin (6 x) = 0

Let:

u = 2 x

sin (u) - sin (2 u) + sin (3 u) = 0

Rewrite using trig identities

- sin (2 u) + sin (3 u) + sin (u)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 2 sin (u) cos (u) + sin (3 u) + sin (u)

sin (3 u) = 3 sin (u) - 4 sin^{3} (u)

sin (3 u)

Rewrite using trig identities

sin (3 u)

Rewrite as

= sin (2 u + u)

Use the Angle Sum identity:

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

= sin (2 u) cos (u) + cos (2 u) sin (u)

Use the Double Angle identity:

sin (2 u) = 2 sin (u) cos (u)

= cos (2 u) sin (u) + cos (u) 2 sin (u) cos (u)

Simplify

cos (2 u) sin (u) + cos (u) \cdot 2 sin (u) cos (u) : sin (u) cos (2 u) + 2 cos^{2} (u) sin (u)

cos (2 u) sin (u) + cos (u) 2 sin (u) cos (u)

cos (u) \cdot 2 sin (u) cos (u) = 2 cos^{2} (u) sin (u)

cos (u) 2 sin (u) cos (u)

Apply exponent rule:

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

cos (u) cos (u) = cos^{1 + 1} (u)

= 2 sin (u) cos^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

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

= sin (u) cos (2 u) + 2 cos^{2} (u) sin (u)

= sin (u) cos (2 u) + 2 cos^{2} (u) sin (u)

= sin (u) cos (2 u) + 2 cos^{2} (u) sin (u)

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

cos^{2} (u) + sin^{2} (u) = 1

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

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

Expand

(1 - 2 sin^{2} (u)) sin (u) + 2 (1 - sin^{2} (u)) sin (u) : - 4 sin^{3} (u) + 3 sin (u)

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

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

Expand

sin (u) (1 - 2 sin^{2} (u)) : sin (u) - 2 sin^{3} (u)

sin (u) (1 - 2 sin^{2} (u))

Apply the distributive law:

a (b - c) = ab - a c

a = sin (u), b = 1, c = 2 sin^{2} (u)

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

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

Simplify

1 \cdot sin (u) - 2 sin^{2} (u) sin (u) : sin (u) - 2 sin^{3} (u)

1 sin (u) - 2 sin^{2} (u) sin (u)

1 \cdot sin (u) = sin (u)

1 sin (u)

Multiply:

1 \cdot sin (u) = sin (u)

= sin (u)

2 sin^{2} (u) sin (u) = 2 sin^{3} (u)

2 sin^{2} (u) sin (u)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 sin^{3} (u)

= sin (u) - 2 sin^{3} (u)

= sin (u) - 2 sin^{3} (u)

= sin (u) - 2 sin^{3} (u) + 2 (1 - sin^{2} (u)) sin (u)

Expand

2 sin (u) (1 - sin^{2} (u)) : 2 sin (u) - 2 sin^{3} (u)

2 sin (u) (1 - sin^{2} (u))

Apply the distributive law:

a (b - c) = ab - a c

a = 2 sin (u), b = 1, c = sin^{2} (u)

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

= 2 \cdot 1 sin (u) - 2 sin^{2} (u) sin (u)

Simplify

2 \cdot 1 \cdot sin (u) - 2 sin^{2} (u) sin (u) : 2 sin (u) - 2 sin^{3} (u)

2 \cdot 1 sin (u) - 2 sin^{2} (u) sin (u)

2 \cdot 1 \cdot sin (u) = 2 sin (u)

2 \cdot 1 sin (u)

Multiply the numbers:

2 \cdot 1 = 2

= 2 sin (u)

2 sin^{2} (u) sin (u) = 2 sin^{3} (u)

2 sin^{2} (u) sin (u)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 sin^{3} (u)

= 2 sin (u) - 2 sin^{3} (u)

= 2 sin (u) - 2 sin^{3} (u)

= sin (u) - 2 sin^{3} (u) + 2 sin (u) - 2 sin^{3} (u)

Simplify

sin (u) - 2 sin^{3} (u) + 2 sin (u) - 2 sin^{3} (u) : - 4 sin^{3} (u) + 3 sin (u)

sin (u) - 2 sin^{3} (u) + 2 sin (u) - 2 sin^{3} (u)

Group like terms

= - 2 sin^{3} (u) - 2 sin^{3} (u) + sin (u) + 2 sin (u)

Add similar elements:

- 2 sin^{3} (u) - 2 sin^{3} (u) = - 4 sin^{3} (u)

= - 4 sin^{3} (u) + sin (u) + 2 sin (u)

Add similar elements:

sin (u) + 2 sin (u) = 3 sin (u)

= - 4 sin^{3} (u) + 3 sin (u)

= - 4 sin^{3} (u) + 3 sin (u)

= - 4 sin^{3} (u) + 3 sin (u)

= 3 sin (u) - 4 sin^{3} (u) + sin (u) - 2 cos (u) sin (u)

Simplify

= 4 sin (u) - 4 sin^{3} (u) - 2 cos (u) sin (u)

4 sin (u) - 4 sin^{3} (u) - 2 cos (u) sin (u) = 0

Factor

4 sin (u) - 4 sin^{3} (u) - 2 cos (u) sin (u) : 2 sin (u) (2 - 2 sin^{2} (u) - cos (u))

4 sin (u) - 4 sin^{3} (u) - 2 cos (u) sin (u)

Apply exponent rule:

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

sin^{3} (u) = sin (u) sin^{2} (u)

= 4 sin (u) - 4 sin (u) sin^{2} (u) - 2 sin (u) cos (u)

Rewrite

- 4

2 \cdot 2

Rewrite

4

2 \cdot 2

= 2 \cdot 2 sin (u) + 2 \cdot 2 sin (u) sin^{2} (u) - 2 sin (u) cos (u)

Factor out common term

2 sin (u)

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

2 sin (u) (2 - 2 sin^{2} (u) - cos (u)) = 0

Solving each part separately

sin (u) = 0 or 2 - 2 sin^{2} (u) - cos (u) = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 - 2 sin^{2} (u) - cos (u) = 0 : u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn, u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

2 - 2 sin^{2} (u) - cos (u) = 0

Rewrite using trig identities

2 - cos (u) - 2 sin^{2} (u)

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

2 - cos (u) - 2 + 2 cos^{2} (u)

Group like terms

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

2 - 2 = 0

= 2 cos^{2} (u) - cos (u)

= 2 cos^{2} (u) - cos (u)

= 2 cos^{2} (u) - cos (u)

- cos (u) + 2 cos^{2} (u) = 0

Solve by substitution

- cos (u) + 2 cos^{2} (u) = 0

Let:

cos (u) = u

- u + 2 u^{2} = 0

- u + 2 u^{2} = 0 : u = \frac{1}{2}, u = 0

- u + 2 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} - u = 0

Solve with the quadratic formula

2 u^{2} - u = 0

Quadratic Equation Formula:

For

a = 2, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(- 1)^{2} - 4 \cdot 2 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 2 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- ( - 1 ) \pm 1}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 1}{2 \cdot 2}, u_{2} = \frac{- ( - 1 ) - 1}{2 \cdot 2}

u = \frac{- ( - 1 ) + 1}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 1 ) + 1}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{1 + 1}{2 \cdot 2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- ( - 1 ) - 1}{2 \cdot 2} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{1 - 1}{2 \cdot 2}

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{4}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = 0

Substitute back

u = cos (u)

cos (u) = \frac{1}{2}, cos (u) = 0

cos (u) = \frac{1}{2}, cos (u) = 0

cos (u) = \frac{1}{2} : u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

cos (u) = \frac{1}{2}

General solutions for

cos (u) = \frac{1}{2}

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}

u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

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

cos (u) = 0

General solutions for

cos (u) = 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}

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

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

Combine all the solutions

u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn, u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn, u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

Substitute back

u = 2 x

2 x = 2 πn : x = πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

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

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} + \frac{2 πn}{2} : \frac{π + 2 πn}{2}

\frac{π}{2} + \frac{2 πn}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

2 x = \frac{π}{3} + 2 πn : x = \frac{π + 6 πn}{6}

2 x = \frac{π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{3}}{2} + \frac{2 πn}{2} : \frac{π + 6 πn}{6}

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{\frac{π}{3} + 2 πn}{2}

Join

\frac{π}{3} + 2 πn : \frac{π + 6 πn}{3}

\frac{π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π + 6 πn}{3}

= \frac{\frac{π + 6 πn}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π + 6 πn}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π + 6 πn}{6}

x = \frac{π + 6 πn}{6}

x = \frac{π + 6 πn}{6}

x = \frac{π + 6 πn}{6}

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π + 6 πn}{6}

2 x = \frac{5 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{3} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2} : \frac{5 π + 6 πn}{6}

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{\frac{5 π}{3} + 2 πn}{2}

Join

\frac{5 π}{3} + 2 πn : \frac{5 π + 6 πn}{3}

\frac{5 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{5 π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{5 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 π + 6 πn}{3}

= \frac{\frac{5 π + 6 πn}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π + 6 πn}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{5 π + 6 πn}{6}

x = \frac{5 π + 6 πn}{6}

x = \frac{5 π + 6 πn}{6}

x = \frac{5 π + 6 πn}{6}

2 x = \frac{π}{2} + 2 πn : x = \frac{π + 4 πn}{4}

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π + 4 πn}{4}

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{\frac{π}{2} + 2 πn}{2}

Join

\frac{π}{2} + 2 πn : \frac{π + 4 πn}{2}

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π + 4 πn}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{4}

2 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π + 4 πn}{4}

2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} : \frac{3 π + 4 πn}{4}

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{\frac{3 π}{2} + 2 πn}{2}

Join

\frac{3 π}{2} + 2 πn : \frac{3 π + 4 πn}{2}

\frac{3 π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π + 4 πn}{2}

= \frac{\frac{3 π + 4 πn}{2}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3 π + 4 πn}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π + 4 πn}{4}

x = \frac{3 π + 4 πn}{4}

x = \frac{3 π + 4 πn}{4}

x = \frac{3 π + 4 πn}{4}

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{π + 6 πn}{6}, x = \frac{5 π + 6 πn}{6}, x = \frac{π + 4 πn}{4}, x = \frac{3 π + 4 πn}{4}

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