What is the general solution for sin(x)=-cos(4x) ?

The general solution for sin(x)=-cos(4x) is x=(pi+4pin)/6 ,x=-(pi+4pin)/(10)

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Popular Trigonometry >

sin(x)=-cos(4x)

Solution

sin (x) = - cos (4 x)

Solution

x = \frac{π + 4 πn}{6}, x = - \frac{π + 4 πn}{10}

Degrees

x = 3 0^{\circ} + 12 0^{\circ} n, x = - 1 8^{\circ} - 7 2^{\circ} n

Solution steps

sin (x) = - cos (4 x)

Multiply by

- 1

- sin (x) = cos (4 x)

Rewrite using trig identities

- sin (x) = cos (4 x)

Use the following identity:

- sin (x) = sin (- x)

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

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (- (x)) = sin (\frac{π}{2} - 4 x)

sin (- (x)) = sin (\frac{π}{2} - 4 x)

Apply trig inverse properties

sin (- (x)) = sin (\frac{π}{2} - 4 x)

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

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

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

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

- (x) = \frac{π}{2} - 4 x + 2 πn

Expand

- (x) : - x

- (x)

Remove parentheses:

(a) = a

= - x

- x = \frac{π}{2} - 4 x + 2 πn

Move

4 x

to the left side

- x = \frac{π}{2} - 4 x + 2 πn

Add

4 x

to both sides

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

Simplify

3 x = \frac{π}{2} + 2 πn

3 x = \frac{π}{2} + 2 πn

Divide both sides by

3

3 x = \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

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

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{3}

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}}{3}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π + 4 πn}{6}

x = \frac{π + 4 πn}{6}

x = \frac{π + 4 πn}{6}

x = \frac{π + 4 πn}{6}

- (x) = π - (\frac{π}{2} - 4 x) + 2 πn : x = - \frac{π + 4 πn}{10}

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

Expand

- (x) : - x

- (x)

Remove parentheses:

(a) = a

= - 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

- x = π - \frac{π}{2} + 4 x + 2 πn

Move

4 x

to the left side

- x = π - \frac{π}{2} + 4 x + 2 πn

Subtract

4 x

from both sides

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

Simplify

- 5 x = π - \frac{π}{2} + 2 πn

- 5 x = π - \frac{π}{2} + 2 πn

Divide both sides by

- 5

- 5 x = π - \frac{π}{2} + 2 πn

Divide both sides by

- 5

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

Simplify

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

Simplify

\frac{- 5 x}{- 5} : x

\frac{- 5 x}{- 5}

Apply the fraction rule:

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

= \frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{π}{- 5} - \frac{\frac{π}{2}}{- 5} + \frac{2 πn}{- 5} : - \frac{π + 4 πn}{10}

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

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{- 5}

Apply the fraction rule:

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

= - \frac{π - \frac{π}{2} + 2 πn}{5}

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}}{5}

Simplify

\frac{\frac{π + 4 πn}{2}}{5} : \frac{π + 4 πn}{10}

\frac{\frac{π + 4 πn}{2}}{5}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{π + 4 πn}{10}

= - \frac{π + 4 πn}{10}

x = - \frac{π + 4 πn}{10}

x = - \frac{π + 4 πn}{10}

x = - \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{6}, x = - \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{6}, x = - \frac{π + 4 πn}{10}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=-cos(4x) ?
The general solution for sin(x)=-cos(4x) is x=(pi+4pin)/6 ,x=-(pi+4pin)/(10)