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

Solution

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

Solution

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

Degrees

x = 13 5^{\circ}, x = 4 5^{\circ}, x = 31 5^{\circ}, x = 22 5^{\circ}

Solution steps

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

Rewrite using trig identities

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

Use the Angle Difference identity:

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

= cos (x - 3 x)

cos (x - 3 x) = 0

General solutions for

cos (x - 3 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}

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

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

Solve

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

x - 3 x = \frac{π}{2} + 2 πn

Add similar elements:

x - 3 x = - 2 x

- 2 x = \frac{π}{2} + 2 πn

Divide both sides by

- 2

- 2 x = \frac{π}{2} + 2 πn

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

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

\frac{\frac{π}{2}}{- 2}

Apply the fraction rule:

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

= - \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

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

= - \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{π}{4}

= - \frac{π}{4} + \frac{2 πn}{- 2}

\frac{2 πn}{- 2} = - πn

\frac{2 πn}{- 2}

Apply the fraction rule:

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

= - \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= - πn

= - \frac{π}{4} - πn

x = - \frac{π}{4} - πn

x = - \frac{π}{4} - πn

x = - \frac{π}{4} - πn

Solve

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

x - 3 x = \frac{3 π}{2} + 2 πn

Add similar elements:

x - 3 x = - 2 x

- 2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

- 2

- 2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

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

\frac{\frac{3 π}{2}}{- 2}

Apply the fraction rule:

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

= - \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

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

= - \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{3 π}{4}

= - \frac{3 π}{4} + \frac{2 πn}{- 2}

\frac{2 πn}{- 2} = - πn

\frac{2 πn}{- 2}

Apply the fraction rule:

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

= - \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= - πn

= - \frac{3 π}{4} - πn

x = - \frac{3 π}{4} - πn

x = - \frac{3 π}{4} - πn

x = - \frac{3 π}{4} - πn

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

Solutions for the range

0 \leq x \leq 2 π

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

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