What is the general solution for cos(4x)+sin(6x)=0 ?

The general solution for cos(4x)+sin(6x)=0 is x=-(pi+4pin)/4 ,x=-(pi+4pin)/(20)

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

Solution

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

Solution

x = - \frac{π + 4 πn}{4}, x = - \frac{π + 4 πn}{20}

Degrees

x = - 4 5^{\circ} - 18 0^{\circ} n, x = - 9^{\circ} - 3 6^{\circ} n

Solution steps

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

Subtract

sin (6 x)

from both sides

cos (4 x) = - sin (6 x)

Rewrite using trig identities

cos (4 x) = - sin (6 x)

Use the following identity:

- sin (x) = sin (- x)

cos (4 x) = sin (- (6 x))

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (\frac{π}{2} - 4 x) = sin (- (6 x))

sin (\frac{π}{2} - 4 x) = sin (- (6 x))

Apply trig inverse properties

sin (\frac{π}{2} - 4 x) = sin (- (6 x))

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

- (6 x) = \frac{π}{2} - 4 x + 2 πn, - (6 x) = π - (\frac{π}{2} - 4 x) + 2 πn

- (6 x) = \frac{π}{2} - 4 x + 2 πn, - (6 x) = π - (\frac{π}{2} - 4 x) + 2 πn

- (6 x) = \frac{π}{2} - 4 x + 2 πn : x = - \frac{π + 4 πn}{4}

- (6 x) = \frac{π}{2} - 4 x + 2 πn

Expand

- (6 x) : - 6 x

- (6 x)

Remove parentheses:

(a) = a

= - 6 x

- 6 x = \frac{π}{2} - 4 x + 2 πn

Move

4 x

to the left side

- 6 x = \frac{π}{2} - 4 x + 2 πn

Add

4 x

to both sides

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

Simplify

- 2 x = \frac{π}{2} + 2 πn

- 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}

Apply the fraction rule:

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

= \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}

Apply the fraction rule:

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

= - \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}

Simplify

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

\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}

= - \frac{π + 4 πn}{4}

x = - \frac{π + 4 πn}{4}

x = - \frac{π + 4 πn}{4}

x = - \frac{π + 4 πn}{4}

- (6 x) = π - (\frac{π}{2} - 4 x) + 2 πn : x = - \frac{π + 4 πn}{20}

- (6 x) = π - (\frac{π}{2} - 4 x) + 2 πn

Expand

- (6 x) : - 6 x

- (6 x)

Remove parentheses:

(a) = a

= - 6 x

Expand

π - (\frac{π}{2} - 4 x) + 2 πn : π - \frac{π}{2} + 4 x + 2 πn

π - (\frac{π}{2} - 4 x) + 2 πn

- (\frac{π}{2} - 4 x) : - \frac{π}{2} + 4 x

- (\frac{π}{2} - 4 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 4 x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + 4 x

= π - \frac{π}{2} + 4 x + 2 πn

- 6 x = π - \frac{π}{2} + 4 x + 2 πn

Move

4 x

to the left side

- 6 x = π - \frac{π}{2} + 4 x + 2 πn

Subtract

4 x

from both sides

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

Simplify

- 10 x = π - \frac{π}{2} + 2 πn

- 10 x = π - \frac{π}{2} + 2 πn

Divide both sides by

- 10

- 10 x = π - \frac{π}{2} + 2 πn

Divide both sides by

- 10

\frac{- 10 x}{- 10} = \frac{π}{- 10} - \frac{\frac{π}{2}}{- 10} + \frac{2 πn}{- 10}

Simplify

\frac{- 10 x}{- 10} = \frac{π}{- 10} - \frac{\frac{π}{2}}{- 10} + \frac{2 πn}{- 10}

Simplify

\frac{- 10 x}{- 10} : x

\frac{- 10 x}{- 10}

Apply the fraction rule:

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

= \frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

\frac{π}{- 10} - \frac{\frac{π}{2}}{- 10} + \frac{2 πn}{- 10} : - \frac{π + 4 πn}{20}

\frac{π}{- 10} - \frac{\frac{π}{2}}{- 10} + \frac{2 πn}{- 10}

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{- 10}

Apply the fraction rule:

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

= - \frac{π - \frac{π}{2} + 2 πn}{10}

Join

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

π - \frac{π}{2} + 2 πn

Convert element to fraction:

π = \frac{π 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 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 - π + 2 πn \cdot 2}{2}

π 2 - π + 2 πn \cdot 2 = π + 4 πn

π 2 - π + 2 πn \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn

= \frac{π + 4 πn}{2}

= - \frac{\frac{π + 4 πn}{2}}{10}

Simplify

\frac{\frac{π + 4 πn}{2}}{10} : \frac{π + 4 πn}{20}

\frac{\frac{π + 4 πn}{2}}{10}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{π + 4 πn}{20}

= - \frac{π + 4 πn}{20}

x = - \frac{π + 4 πn}{20}

x = - \frac{π + 4 πn}{20}

x = - \frac{π + 4 πn}{20}

x = - \frac{π + 4 πn}{4}, x = - \frac{π + 4 πn}{20}

x = - \frac{π + 4 πn}{4}, x = - \frac{π + 4 πn}{20}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(4x)+sin(6x)=0 ?
The general solution for cos(4x)+sin(6x)=0 is x=-(pi+4pin)/4 ,x=-(pi+4pin)/(20)