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cos(3x)cos(9x)+sin(3x)sin(9x)=0,(0,2pi)

Solution

cos (3 x) cos (9 x) + sin (3 x) sin (9 x) = 0, (0, 2 π)

Solution

x = \frac{π}{4}, x = \frac{π}{12}, x = \frac{7 π}{12}, x = \frac{5 π}{12}, x = \frac{11 π}{12}, x = \frac{3 π}{4}, x = \frac{5 π}{4}, x = \frac{13 π}{12}

Degrees

x = 4 5^{\circ}, x = 1 5^{\circ}, x = 10 5^{\circ}, x = 7 5^{\circ}, x = 16 5^{\circ}, x = 13 5^{\circ}, x = 22 5^{\circ}, x = 19 5^{\circ}

Solution steps

cos (3 x) cos (9 x) + sin (3 x) sin (9 x) = 0, (0, 2 π)

Rewrite using trig identities

cos (3 x) cos (9 x) + sin (3 x) sin (9 x)

Use the Angle Difference identity:

cos (s) cos (t) + sin (s) sin (t) = cos (s - t)

= cos (3 x - 9 x)

cos (3 x - 9 x) = 0

General solutions for

cos (3 x - 9 x) = 0

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 - 9 x = \frac{π}{2} + 2 πn, 3 x - 9 x = \frac{3 π}{2} + 2 πn

3 x - 9 x = \frac{π}{2} + 2 πn, 3 x - 9 x = \frac{3 π}{2} + 2 πn

Solve

3 x - 9 x = \frac{π}{2} + 2 πn : x = - \frac{π}{12} - \frac{πn}{3}

3 x - 9 x = \frac{π}{2} + 2 πn

Add similar elements:

3 x - 9 x = - 6 x

- 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}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} : - \frac{π}{12} - \frac{πn}{3}

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

\frac{\frac{π}{2}}{- 6} = - \frac{π}{12}

\frac{\frac{π}{2}}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{π}{2}}{6}

Apply the fraction rule:

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

\frac{\frac{π}{2}}{6} = \frac{π}{2 \cdot 6}

= - \frac{π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= - \frac{π}{12}

= - \frac{π}{12} + \frac{2 πn}{- 6}

\frac{2 πn}{- 6} = - \frac{πn}{3}

\frac{2 πn}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 πn}{6}

Cancel the common factor:

2

= - \frac{πn}{3}

= - \frac{π}{12} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}

Solve

3 x - 9 x = \frac{3 π}{2} + 2 πn : x = - \frac{π}{4} - \frac{πn}{3}

3 x - 9 x = \frac{3 π}{2} + 2 πn

Add similar elements:

3 x - 9 x = - 6 x

- 6 x = \frac{3 π}{2} + 2 πn

Divide both sides by

- 6

- 6 x = \frac{3 π}{2} + 2 πn

Divide both sides by

- 6

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

Simplify

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

Simplify

\frac{- 6 x}{- 6} : x

\frac{- 6 x}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

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

\frac{\frac{3 π}{2}}{- 6} + \frac{2 πn}{- 6}

\frac{\frac{3 π}{2}}{- 6} = - \frac{π}{4}

\frac{\frac{3 π}{2}}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{3 π}{2}}{6}

Apply the fraction rule:

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

\frac{\frac{3 π}{2}}{6} = \frac{3 π}{2 \cdot 6}

= - \frac{3 π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= - \frac{3 π}{12}

Cancel the common factor:

3

= - \frac{π}{4}

= - \frac{π}{4} + \frac{2 πn}{- 6}

\frac{2 πn}{- 6} = - \frac{πn}{3}

\frac{2 πn}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 πn}{6}

Cancel the common factor:

2

= - \frac{πn}{3}

= - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}, x = - \frac{π}{4} - \frac{πn}{3}

Solutions for the range

(0, 2 π)

x = \frac{π}{4}, x = \frac{π}{12}, x = \frac{7 π}{12}, x = \frac{5 π}{12}, x = \frac{11 π}{12}, x = \frac{3 π}{4}, x = \frac{5 π}{4}, x = \frac{13 π}{12}

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