What is the general solution for 2sin(6x)sin(3x)=1-cos(9x) ?

The general solution for 2sin(6x)sin(3x)=1-cos(9x) is x=(2pin)/3

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Popular Trigonometry >

2sin(6x)sin(3x)=1-cos(9x)

Solution

2 sin (6 x) sin (3 x) = 1 - cos (9 x)

Solution

x = \frac{2 πn}{3}

Degrees

x = 0^{\circ} + 12 0^{\circ} n

Solution steps

2 sin (6 x) sin (3 x) = 1 - cos (9 x)

Subtract

1 - cos (9 x)

from both sides

2 sin (6 x) sin (3 x) - 1 + cos (9 x) = 0

Rewrite using trig identities

- 1 + cos (9 x) + 2 sin (3 x) sin (6 x)

Use the Product to Sum identity:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= - 1 + cos (9 x) + 2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x))

Simplify

- 1 + cos (9 x) + 2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x)) : cos (3 x) - 1

- 1 + cos (9 x) + 2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x))

2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x)) = cos (3 x) - cos (9 x)

2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x))

Multiply fractions:

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

= \frac{1 \cdot 2 ( cos ( 3 x - 6 x ) - cos ( 3 x + 6 x ) )}{2}

Cancel the common factor:

2

= 1 \cdot (cos (3 x - 6 x) - cos (3 x + 6 x))

Add similar elements:

3 x - 6 x = - 3 x

= 1 \cdot (cos (- 3 x) - cos (3 x + 6 x))

Add similar elements:

3 x + 6 x = 9 x

= 1 \cdot (cos (- 3 x) - cos (9 x))

Refine

= cos (- 3 x) - cos (9 x)

Use the negative angle identity:

cos (- x) = cos (x)

= cos (3 x) - cos (9 x)

= - 1 + cos (9 x) + cos (3 x) - cos (9 x)

Group like terms

= cos (9 x) + cos (3 x) - cos (9 x) - 1

Add similar elements:

cos (9 x) - cos (9 x) = 0

= cos (3 x) - 1

= cos (3 x) - 1

- 1 + cos (3 x) = 0

Move

1

to the right side

- 1 + cos (3 x) = 0

Add

1

to both sides

- 1 + cos (3 x) + 1 = 0 + 1

Simplify

cos (3 x) = 1

cos (3 x) = 1

General solutions for

cos (3 x) = 1

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

3 x = 0 + 2 πn

3 x = 0 + 2 πn

Solve

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(6x)sin(3x)=1-cos(9x) ?
The general solution for 2sin(6x)sin(3x)=1-cos(9x) is x=(2pin)/3