What is the general solution for sin(5x)=cos(x) ?

The general solution for sin(5x)=cos(x) is x=(pi+4pin}{12},x=\frac{pi+4pin)/8

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sin(5x)=cos(x)

Solution

sin (5 x) = cos (x)

Solution

x = \frac{π + 4 πn}{12}, x = \frac{π + 4 πn}{8}

Degrees

x = 1 5^{\circ} + 6 0^{\circ} n, x = 22. 5^{\circ} + 9 0^{\circ} n

Solution steps

sin (5 x) = cos (x)

Rewrite using trig identities

sin (5 x) = cos (x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x) = sin (\frac{π}{2} - x)

sin (5 x) = sin (\frac{π}{2} - x)

Apply trig inverse properties

sin (5 x) = sin (\frac{π}{2} - x)

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

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

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

5 x = \frac{π}{2} - x + 2 πn : x = \frac{π + 4 πn}{12}

5 x = \frac{π}{2} - x + 2 πn

Move

x

to the left side

5 x = \frac{π}{2} - x + 2 πn

Add

x

to both sides

5 x + x = \frac{π}{2} - x + 2 πn + x

Simplify

6 x = \frac{π}{2} + 2 πn

6 x = \frac{π}{2} + 2 πn

Divide both sides by

6

6 x = \frac{π}{2} + 2 πn

Divide both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6} : \frac{π + 4 πn}{12}

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

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{6}

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}}{6}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π + 4 πn}{12}

x = \frac{π + 4 πn}{12}

x = \frac{π + 4 πn}{12}

x = \frac{π + 4 πn}{12}

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

5 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

5 x = π - \frac{π}{2} + x + 2 πn

Move

x

to the left side

5 x = π - \frac{π}{2} + x + 2 πn

Subtract

x

from both sides

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

Simplify

4 x = π - \frac{π}{2} + 2 πn

4 x = π - \frac{π}{2} + 2 πn

Divide both sides by

4

4 x = π - \frac{π}{2} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

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

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{4}

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}}{4}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{12}, x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{12}, x = \frac{π + 4 πn}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x)=cos(x) ?
The general solution for sin(5x)=cos(x) is x=(pi+4pin}{12},x=\frac{pi+4pin)/8