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Popular >

sin(4x)cos(6x)-cos(4x)sin(6x)=-0.75

Solution

sin (4 x) cos (6 x) - cos (4 x) sin (6 x) = - 0.75

Solution

x = \frac{0.84806 \dots}{2} - πn, x = - \frac{π}{2} - \frac{0.84806 \dots}{2} - πn

Degrees

x = 24.29518 \dots^{\circ} - 18 0^{\circ} n, x = - 114.29518 \dots^{\circ} - 18 0^{\circ} n

Solution steps

sin (4 x) cos (6 x) - cos (4 x) sin (6 x) = - 0.75

Rewrite using trig identities

sin (4 x) cos (6 x) - cos (4 x) sin (6 x)

Use the Angle Difference identity:

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

= sin (4 x - 6 x)

sin (4 x - 6 x) = - 0.75

Apply trig inverse properties

sin (4 x - 6 x) = - 0.75

General solutions for

sin (4 x - 6 x) = - 0.75

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

4 x - 6 x = arcsin (- 0.75) + 2 πn, 4 x - 6 x = π + arcsin (0.75) + 2 πn

4 x - 6 x = arcsin (- 0.75) + 2 πn, 4 x - 6 x = π + arcsin (0.75) + 2 πn

Solve

4 x - 6 x = arcsin (- 0.75) + 2 πn : x = \frac{arcsin ( \frac{3}{4} )}{2} - πn

4 x - 6 x = arcsin (- 0.75) + 2 πn

Simplify

4 x - 6 x : - 2 x

4 x - 6 x

Add similar elements:

4 x - 6 x = - 2 x

= - 2 x

Simplify

arcsin (- 0.75) + 2 πn : - arcsin (\frac{3}{4}) + 2 πn

arcsin (- 0.75) + 2 πn

arcsin (- 0.75) = - arcsin (\frac{3}{4})

arcsin (- 0.75)

= arcsin (- \frac{3}{4})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{4}) = - arcsin (\frac{3}{4})

= - arcsin (\frac{3}{4})

= - arcsin (\frac{3}{4}) + 2 πn

- 2 x = - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

- 2

- 2 x = - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

- 2

\frac{- 2 x}{- 2} = - \frac{arcsin ( \frac{3}{4} )}{- 2} + \frac{2 πn}{- 2}

Simplify

\frac{- 2 x}{- 2} = - \frac{arcsin ( \frac{3}{4} )}{- 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{arcsin ( \frac{3}{4} )}{- 2} + \frac{2 πn}{- 2} : \frac{arcsin ( \frac{3}{4} )}{2} - πn

- \frac{arcsin ( \frac{3}{4} )}{- 2} + \frac{2 πn}{- 2}

\frac{arcsin ( \frac{3}{4} )}{- 2} = - \frac{arcsin ( \frac{3}{4} )}{2}

\frac{arcsin ( \frac{3}{4} )}{- 2}

Apply the fraction rule:

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

= - \frac{arcsin ( \frac{3}{4} )}{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{arcsin ( \frac{3}{4} )}{2}) - πn

Apply rule

- (- a) = a

= \frac{arcsin ( \frac{3}{4} )}{2} - πn

x = \frac{arcsin ( \frac{3}{4} )}{2} - πn

x = \frac{arcsin ( \frac{3}{4} )}{2} - πn

x = \frac{arcsin ( \frac{3}{4} )}{2} - πn

Solve

4 x - 6 x = π + arcsin (0.75) + 2 πn : x = - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} - πn

4 x - 6 x = π + arcsin (0.75) + 2 πn

Add similar elements:

4 x - 6 x = - 2 x

- 2 x = π + arcsin (0.75) + 2 πn

Divide both sides by

- 2

- 2 x = π + arcsin (0.75) + 2 πn

Divide both sides by

- 2

\frac{- 2 x}{- 2} = \frac{π}{- 2} + \frac{arcsin ( 0.75 )}{- 2} + \frac{2 πn}{- 2}

Simplify

\frac{- 2 x}{- 2} = \frac{π}{- 2} + \frac{arcsin ( 0.75 )}{- 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{π}{- 2} + \frac{arcsin ( 0.75 )}{- 2} + \frac{2 πn}{- 2} : - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} - πn

\frac{π}{- 2} + \frac{arcsin ( 0.75 )}{- 2} + \frac{2 πn}{- 2}

Apply the fraction rule:

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

= - \frac{π}{2} + \frac{arcsin ( 0.75 )}{- 2} + \frac{2 πn}{- 2}

Apply the fraction rule:

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

= - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} + \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}{2} - \frac{arcsin ( 0.75 )}{2} - πn

x = - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} - πn

x = - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} - πn

x = - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} - πn

x = \frac{arcsin ( \frac{3}{4} )}{2} - πn, x = - \frac{π}{2} - \frac{arcsin ( 0.75 )}{2} - πn

Show solutions in decimal form

x = \frac{0.84806 \dots}{2} - πn, x = - \frac{π}{2} - \frac{0.84806 \dots}{2} - πn

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