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Popular >

sin(6x)cos(9x)-cos(6x)sin(9x)=-0.4

Solution

sin (6 x) cos (9 x) - cos (6 x) sin (9 x) = - 0.4

Solution

x = \frac{0.41151 \dots}{3} - \frac{2 πn}{3}, x = - \frac{π}{3} - \frac{0.41151 \dots}{3} - \frac{2 πn}{3}

Degrees

x = 7.85939 \dots^{\circ} - 12 0^{\circ} n, x = - 67.85939 \dots^{\circ} - 12 0^{\circ} n

Solution steps

sin (6 x) cos (9 x) - cos (6 x) sin (9 x) = - 0.4

Rewrite using trig identities

sin (6 x) cos (9 x) - cos (6 x) sin (9 x)

Use the Angle Difference identity:

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

= sin (6 x - 9 x)

sin (6 x - 9 x) = - 0.4

Apply trig inverse properties

sin (6 x - 9 x) = - 0.4

General solutions for

sin (6 x - 9 x) = - 0.4

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

6 x - 9 x = arcsin (- 0.4) + 2 πn, 6 x - 9 x = π + arcsin (0.4) + 2 πn

6 x - 9 x = arcsin (- 0.4) + 2 πn, 6 x - 9 x = π + arcsin (0.4) + 2 πn

Solve

6 x - 9 x = arcsin (- 0.4) + 2 πn : x = \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}

6 x - 9 x = arcsin (- 0.4) + 2 πn

Simplify

6 x - 9 x : - 3 x

6 x - 9 x

Add similar elements:

6 x - 9 x = - 3 x

= - 3 x

Simplify

arcsin (- 0.4) + 2 πn : - arcsin (\frac{2}{5}) + 2 πn

arcsin (- 0.4) + 2 πn

arcsin (- 0.4) = - arcsin (\frac{2}{5})

arcsin (- 0.4)

= arcsin (- \frac{2}{5})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{5}) = - arcsin (\frac{2}{5})

= - arcsin (\frac{2}{5})

= - arcsin (\frac{2}{5}) + 2 πn

- 3 x = - arcsin (\frac{2}{5}) + 2 πn

Divide both sides by

- 3

- 3 x = - arcsin (\frac{2}{5}) + 2 πn

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

- \frac{arcsin ( \frac{2}{5} )}{- 3} + \frac{2 πn}{- 3} : \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}

- \frac{arcsin ( \frac{2}{5} )}{- 3} + \frac{2 πn}{- 3}

Apply the fraction rule:

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

= - (- \frac{arcsin ( \frac{2}{5} )}{3}) + \frac{2 πn}{- 3}

Apply the fraction rule:

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

= - (- \frac{arcsin ( \frac{2}{5} )}{3}) - \frac{2 πn}{3}

Apply rule

- (- a) = a

= \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}

x = \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}

x = \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}

x = \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}

Solve

6 x - 9 x = π + arcsin (0.4) + 2 πn : x = - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

6 x - 9 x = π + arcsin (0.4) + 2 πn

Add similar elements:

6 x - 9 x = - 3 x

- 3 x = π + arcsin (0.4) + 2 πn

Divide both sides by

- 3

- 3 x = π + arcsin (0.4) + 2 πn

Divide both sides by

- 3

\frac{- 3 x}{- 3} = \frac{π}{- 3} + \frac{arcsin ( 0.4 )}{- 3} + \frac{2 πn}{- 3}

Simplify

\frac{- 3 x}{- 3} = \frac{π}{- 3} + \frac{arcsin ( 0.4 )}{- 3} + \frac{2 πn}{- 3}

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{- 3} + \frac{arcsin ( 0.4 )}{- 3} + \frac{2 πn}{- 3} : - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

\frac{π}{- 3} + \frac{arcsin ( 0.4 )}{- 3} + \frac{2 πn}{- 3}

Apply the fraction rule:

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

= - \frac{π}{3} + \frac{arcsin ( 0.4 )}{- 3} + \frac{2 πn}{- 3}

Apply the fraction rule:

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

= - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} + \frac{2 πn}{- 3}

Apply the fraction rule:

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

= - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

x = - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

x = - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

x = - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

x = \frac{arcsin ( \frac{2}{5} )}{3} - \frac{2 πn}{3}, x = - \frac{π}{3} - \frac{arcsin ( 0.4 )}{3} - \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.41151 \dots}{3} - \frac{2 πn}{3}, x = - \frac{π}{3} - \frac{0.41151 \dots}{3} - \frac{2 πn}{3}

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