What is the general solution for sin(5x)=cos(3x) ?

The general solution for sin(5x)=cos(3x) is x=(pi+4pin}{16},x=\frac{pi+4pin)/4

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sin(5x)=cos(3x)

Solution

sin (5 x) = cos (3 x)

Solution

x = \frac{π + 4 πn}{16}, x = \frac{π + 4 πn}{4}

Degrees

x = 11.2 5^{\circ} + 4 5^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (5 x) = cos (3 x)

Rewrite using trig identities

sin (5 x) = cos (3 x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x) = sin (\frac{π}{2} - 3 x)

sin (5 x) = sin (\frac{π}{2} - 3 x)

Apply trig inverse properties

sin (5 x) = sin (\frac{π}{2} - 3 x)

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

5 x = \frac{π}{2} - 3 x + 2 πn, 5 x = π - (\frac{π}{2} - 3 x) + 2 πn

5 x = \frac{π}{2} - 3 x + 2 πn, 5 x = π - (\frac{π}{2} - 3 x) + 2 πn

5 x = \frac{π}{2} - 3 x + 2 πn : x = \frac{π + 4 πn}{16}

5 x = \frac{π}{2} - 3 x + 2 πn

Move

3 x

to the left side

5 x = \frac{π}{2} - 3 x + 2 πn

Add

3 x

to both sides

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

Simplify

8 x = \frac{π}{2} + 2 πn

8 x = \frac{π}{2} + 2 πn

Divide both sides by

8

8 x = \frac{π}{2} + 2 πn

Divide both sides by

8

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

Simplify

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

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8} : \frac{π + 4 πn}{16}

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

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{8}

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}}{8}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{16}

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

5 x = π - (\frac{π}{2} - 3 x) + 2 πn

Expand

π - (\frac{π}{2} - 3 x) + 2 πn : π - \frac{π}{2} + 3 x + 2 πn

π - (\frac{π}{2} - 3 x) + 2 πn

- (\frac{π}{2} - 3 x) : - \frac{π}{2} + 3 x

- (\frac{π}{2} - 3 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x

= π - \frac{π}{2} + 3 x + 2 πn

5 x = π - \frac{π}{2} + 3 x + 2 πn

Move

3 x

to the left side

5 x = π - \frac{π}{2} + 3 x + 2 πn

Subtract

3 x

from both sides

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

Simplify

2 x = π - \frac{π}{2} + 2 πn

2 x = π - \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = π - \frac{π}{2} + 2 πn

Divide both sides by

2

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

Simplify

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

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

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{2}

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}}{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}

x = \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{16}, x = \frac{π + 4 πn}{4}

x = \frac{π + 4 πn}{16}, x = \frac{π + 4 πn}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x)=cos(3x) ?
The general solution for sin(5x)=cos(3x) is x=(pi+4pin}{16},x=\frac{pi+4pin)/4