English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(2x)+cos(4x)+cos(6x)=0

Solution

cos (2 x) + cos (4 x) + cos (6 x) = 0

Solution

x = \frac{π ( 3 n + 1 )}{3}, x = \frac{π ( 3 n + 2 )}{3}, x = \frac{3 π + 8 πn}{8}, x = \frac{5 π + 8 πn}{8}, x = \frac{π + 8 πn}{8}, x = \frac{7 π + 8 πn}{8}

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n, x = 112. 5^{\circ} + 18 0^{\circ} n, x = 22. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n

Solution steps

cos (2 x) + cos (4 x) + cos (6 x) = 0

Let:

u = 2 x

cos (u) + cos (2 u) + cos (3 u) = 0

Rewrite using trig identities

cos (2 u) + cos (3 u) + cos (u)

Use the Double Angle identity:

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

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

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

cos (3 u)

Rewrite using trig identities

cos (3 u)

Rewrite as

= cos (2 u + u)

Use the Angle Sum identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

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

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

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

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (u)

1 \cdot cos (u) = cos (u)

1 cos (u)

Multiply:

1 \cdot cos (u) = cos (u)

= cos (u)

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

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

2 \cdot 1 cos (u)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (u)

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

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

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (u)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

Add similar elements:

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

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

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

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

= - 1 + 4 cos^{3} (u) - 3 cos (u) + cos (u) + 2 cos^{2} (u)

Simplify

= - 1 + 4 cos^{3} (u) - 2 cos (u) + 2 cos^{2} (u)

- 1 - 2 cos (u) + 2 cos^{2} (u) + 4 cos^{3} (u) = 0

Solve by substitution

- 1 - 2 cos (u) + 2 cos^{2} (u) + 4 cos^{3} (u) = 0

Let:

cos (u) = u

- 1 - 2 u + 2 u^{2} + 4 u^{3} = 0

- 1 - 2 u + 2 u^{2} + 4 u^{3} = 0 : u = - \frac{1}{2}, u = - \frac{2}{2}, u = \frac{2}{2}

- 1 - 2 u + 2 u^{2} + 4 u^{3} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

4 u^{3} + 2 u^{2} - 2 u - 1 = 0

Factor

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

4 u^{3} + 2 u^{2} - 2 u - 1

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

Factor out

- 1

from

- 2 u - 1 : - (2 u + 1)

- 2 u - 1

Factor out common term

- 1

= - (2 u + 1)

Factor out

2 u^{2}

from

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

4 u^{3} + 2 u^{2}

Apply exponent rule:

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

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

= 4 u u^{2} + 2 u^{2}

Rewrite

4

2 \cdot 2

= 2 \cdot 2 u u^{2} + 2 u^{2}

Factor out common term

2 u^{2}

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

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

Factor out common term

2 u + 1

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

Factor

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

2 u^{2} - 1

Rewrite

2 u^{2} - 1

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

2 u^{2} - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Rewrite

1

1^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

= (2 u + 1) (2 u - 1)

= (2 u + 1) (2 u + 1) (2 u - 1)

(2 u + 1) (2 u + 1) (2 u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

2 u + 1 = 0 or 2 u + 1 = 0 or 2 u - 1 = 0

Solve

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

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

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

Simplify

u = - \frac{1}{2}

u = - \frac{1}{2}

Solve

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

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

Solve

2 u - 1 = 0 : u = \frac{2}{2}

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

The solutions are

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

Substitute back

u = cos (u)

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

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

cos (u) = - \frac{1}{2} : u = \frac{2 π}{3} + 2 πn, u = \frac{4 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{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

cos (u) = - \frac{2}{2} : u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

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

General solutions for

cos (u) = - \frac{2}{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{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

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

cos (u) = \frac{2}{2} : u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

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

General solutions for

cos (u) = \frac{2}{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}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

Combine all the solutions

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn, u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn, u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

Substitute back

u = 2 x

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

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

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

Apply rule

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

= \frac{\frac{2 π}{3} + 2 πn}{2}

Join

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

\frac{2 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{2 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{2 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2 π + 6 πn}{3}

= \frac{\frac{2 π + 6 πn}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π + 6 πn}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π + 6 πn}{6}

Factor

2 π + 6 πn : 2 π (1 + 3 n)

2 π + 6 πn

Rewrite as

= 1 \cdot 2 π + 3 \cdot 2 πn

Factor out common term

2 π

= 2 π (1 + 3 n)

= \frac{2 π ( 1 + 3 n )}{6}

Cancel the common factor:

2

= \frac{π ( 3 n + 1 )}{3}

x = \frac{π ( 3 n + 1 )}{3}

x = \frac{π ( 3 n + 1 )}{3}

x = \frac{π ( 3 n + 1 )}{3}

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

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

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

Apply rule

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

= \frac{\frac{4 π}{3} + 2 πn}{2}

Join

\frac{4 π}{3} + 2 πn : \frac{4 π + 6 πn}{3}

\frac{4 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{4 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{4 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4 π + 6 πn}{3}

= \frac{\frac{4 π + 6 πn}{3}}{2}

Apply the fraction rule:

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

= \frac{4 π + 6 πn}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{4 π + 6 πn}{6}

Factor

4 π + 6 πn : 2 π (2 + 3 n)

4 π + 6 πn

Rewrite as

= 2 \cdot 2 π + 3 \cdot 2 πn

Factor out common term

2 π

= 2 π (2 + 3 n)

= \frac{2 π ( 2 + 3 n )}{6}

Cancel the common factor:

2

= \frac{π ( 3 n + 2 )}{3}

x = \frac{π ( 3 n + 2 )}{3}

x = \frac{π ( 3 n + 2 )}{3}

x = \frac{π ( 3 n + 2 )}{3}

2 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π + 8 πn}{8}

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{3 π}{4}}{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}{4}}{2} + \frac{2 πn}{2} : \frac{3 π + 8 πn}{8}

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

Apply rule

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

= \frac{\frac{3 π}{4} + 2 πn}{2}

Join

\frac{3 π}{4} + 2 πn : \frac{3 π + 8 πn}{4}

\frac{3 π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{3 π}{4} + \frac{2 πn \cdot 4}{4}

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 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 π + 8 πn}{4}

= \frac{\frac{3 π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{3 π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{3 π + 8 πn}{8}

x = \frac{3 π + 8 πn}{8}

x = \frac{3 π + 8 πn}{8}

x = \frac{3 π + 8 πn}{8}

2 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π + 8 πn}{8}

2 x = \frac{5 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{4}}{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}{4}}{2} + \frac{2 πn}{2} : \frac{5 π + 8 πn}{8}

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

Apply rule

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

= \frac{\frac{5 π}{4} + 2 πn}{2}

Join

\frac{5 π}{4} + 2 πn : \frac{5 π + 8 πn}{4}

\frac{5 π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{5 π}{4} + \frac{2 πn \cdot 4}{4}

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 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{5 π + 8 πn}{4}

= \frac{\frac{5 π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{5 π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{5 π + 8 πn}{8}

x = \frac{5 π + 8 πn}{8}

x = \frac{5 π + 8 πn}{8}

x = \frac{5 π + 8 πn}{8}

2 x = \frac{π}{4} + 2 πn : x = \frac{π + 8 πn}{8}

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{π}{4}}{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}{4}}{2} + \frac{2 πn}{2} : \frac{π + 8 πn}{8}

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

Apply rule

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

= \frac{\frac{π}{4} + 2 πn}{2}

Join

\frac{π}{4} + 2 πn : \frac{π + 8 πn}{4}

\frac{π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{π}{4} + \frac{2 πn \cdot 4}{4}

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 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 8 πn}{4}

= \frac{\frac{π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π + 8 πn}{8}

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{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{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π + 8 πn}{8}

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Apply rule

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

= \frac{\frac{7 π}{4} + 2 πn}{2}

Join

\frac{7 π}{4} + 2 πn : \frac{7 π + 8 πn}{4}

\frac{7 π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{7 π}{4} + \frac{2 πn \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{7 π + 8 πn}{4}

= \frac{\frac{7 π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{7 π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{π ( 3 n + 1 )}{3}, x = \frac{π ( 3 n + 2 )}{3}, x = \frac{3 π + 8 πn}{8}, x = \frac{5 π + 8 πn}{8}, x = \frac{π + 8 πn}{8}, x = \frac{7 π + 8 πn}{8}

Popular Examples

sqrt(2)cos(x)sin(x)-cos(x)=0

2 cos (x) sin (x) - cos (x) = 0

6sin(x)=6sin(2x)

6 sin (x) = 6 sin (2 x)

sin(5x)+sin(x)=sqrt(3)cos(2x)

sin (5 x) + sin (x) = 3 cos (2 x)

0=-5sin(2x)

0 = - 5 sin (2 x)

3sin(2x)-1=0

3 sin (2 x) - 1 = 0