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

The general solution for cos(x)=sin(4x) is x=(pi+4pin}{10},x=\frac{pi+4pin)/6

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cos(x)=sin(4x)

Solution

cos (x) = sin (4 x)

Solution

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

Degrees

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

Solution steps

cos (x) = sin (4 x)

Rewrite using trig identities

cos (x) = sin (4 x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (x) = sin (\frac{π}{2} - x)

cos (x) = sin (\frac{π}{2} - x)

Apply trig inverse properties

cos (x) = sin (\frac{π}{2} - x)

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

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

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

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

4 x = \frac{π}{2} - x + 2 πn

Move

x

to the left side

4 x = \frac{π}{2} - x + 2 πn

Add

x

to both sides

4 x + x = \frac{π}{2} - x + 2 πn + x

Simplify

5 x = \frac{π}{2} + 2 πn

5 x = \frac{π}{2} + 2 πn

Divide both sides by

5

5 x = \frac{π}{2} + 2 πn

Divide both sides by

5

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

Simplify

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

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

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

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

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{5}

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}}{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}

x = \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{10}

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

4 x = π - (\frac{π}{2} - x) + 2 πn

Expand

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

π - (\frac{π}{2} - x) + 2 πn

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

- (\frac{π}{2} - x)

Distribute parentheses

= - (\frac{π}{2}) - (- x)

Apply minus-plus rules

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

= - \frac{π}{2} + x

= π - \frac{π}{2} + x + 2 πn

4 x = π - \frac{π}{2} + x + 2 πn

Move

x

to the left side

4 x = π - \frac{π}{2} + x + 2 πn

Subtract

x

from both sides

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

Simplify

3 x = π - \frac{π}{2} + 2 πn

3 x = π - \frac{π}{2} + 2 πn

Divide both sides by

3

3 x = π - \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{3}

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

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

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Frequently Asked Questions (FAQ)

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