What is the general solution for (cos(2x))/(sin(3x)-sin(x))-1=0 ?

The general solution for (cos(2x))/(sin(3x)-sin(x))-1=0 is x= pi/6+2pin,x=(5pi)/6+2pin

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(cos(2x))/(sin(3x)-sin(x))-1=0

Solution

\frac{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1 = 0

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1 = 0

Rewrite using trig identities

\frac{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= \frac{cos ( 2 x )}{2 sin ( \frac{3 x - x}{2} ) cos ( \frac{3 x + x}{2} )} - 1

\frac{cos ( 2 x )}{2 sin ( \frac{3 x - x}{2} ) cos ( \frac{3 x + x}{2} )} = \frac{1}{2 sin ( x )}

\frac{cos ( 2 x )}{2 sin ( \frac{3 x - x}{2} ) cos ( \frac{3 x + x}{2} )}

2 sin (\frac{3 x - x}{2}) cos (\frac{3 x + x}{2}) = 2 sin (\frac{2 x}{2}) cos (\frac{4 x}{2})

2 sin (\frac{3 x - x}{2}) cos (\frac{3 x + x}{2})

Add similar elements:

3 x - x = 2 x

= 2 sin (\frac{2 x}{2}) cos (\frac{3 x + x}{2})

Add similar elements:

3 x + x = 4 x

= 2 sin (\frac{2 x}{2}) cos (\frac{4 x}{2})

= \frac{cos ( 2 x )}{2 sin ( \frac{2 x}{2} ) cos ( \frac{4 x}{2} )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{cos ( 2 x )}{2 sin ( x ) cos ( \frac{4 x}{2} )}

Divide the numbers:

\frac{4}{2} = 2

= \frac{cos ( 2 x )}{2 sin ( x ) cos ( 2 x )}

Cancel the common factor:

cos (2 x)

= \frac{1}{2 sin ( x )}

= \frac{1}{2 sin ( x )} - 1

\frac{1}{2 sin ( x )} - 1 = 0

Multiply both sides by

2 sin (x)

\frac{1}{2 sin ( x )} - 1 = 0

Multiply both sides by

2 sin (x)

\frac{1}{2 sin ( x )} \cdot 2 sin (x) - 1 \cdot 2 sin (x) = 0 \cdot 2 sin (x)

Simplify

\frac{1}{2 sin ( x )} \cdot 2 sin (x) - 1 \cdot 2 sin (x) = 0 \cdot 2 sin (x)

Simplify

\frac{1}{2 sin ( x )} \cdot 2 sin (x) : 1

\frac{1}{2 sin ( x )} \cdot 2 sin (x)

Multiply fractions:

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

= \frac{1 \cdot 2 sin ( x )}{2 sin ( x )}

Cancel the common factor:

2

= \frac{1 \cdot sin ( x )}{sin ( x )}

Cancel the common factor:

sin (x)

= 1

Simplify

- 1 \cdot 2 sin (x) : - 2 sin (x)

- 1 \cdot 2 sin (x)

Multiply the numbers:

1 \cdot 2 = 2

= - 2 sin (x)

Simplify

0 \cdot 2 sin (x) : 0

0 \cdot 2 sin (x)

Apply rule

0 \cdot a = 0

= 0

1 - 2 sin (x) = 0

1 - 2 sin (x) = 0

1 - 2 sin (x) = 0

Move

1

to the right side

1 - 2 sin (x) = 0

Subtract

1

from both sides

1 - 2 sin (x) - 1 = 0 - 1

Simplify

- 2 sin (x) = - 1

- 2 sin (x) = - 1

Divide both sides by

- 2

- 2 sin (x) = - 1

Divide both sides by

- 2

\frac{- 2 sin ( x )}{- 2} = \frac{- 1}{- 2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{1}{2 sin ( x )} - 1

and compare to zero

Solve

2 sin (x) = 0 : sin (x) = 0

2 sin (x) = 0

Divide both sides by

2

2 sin (x) = 0

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{0}{2}

Simplify

sin (x) = 0

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (cos(2x))/(sin(3x)-sin(x))-1=0 ?
The general solution for (cos(2x))/(sin(3x)-sin(x))-1=0 is x= pi/6+2pin,x=(5pi)/6+2pin