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Popular >

cos(3/2 pi-x)+sin(4x)=0

Solution

cos (\frac{3}{2} π - x) + sin (4 x) = 0

Solution

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Degrees

x = 3 6^{\circ} + 14 4^{\circ} n, x = 10 8^{\circ} + 14 4^{\circ} n, x = 0^{\circ} + 24 0^{\circ} n, x = 12 0^{\circ} + 24 0^{\circ} n

Solution steps

cos (\frac{3}{2} π - x) + sin (4 x) = 0

Rewrite using trig identities

cos (\frac{3}{2} π - x) + sin (4 x) = 0

Rewrite using trig identities

cos (\frac{3}{2} π - x)

Use the Angle Difference identity:

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

= cos (\frac{3}{2} π) cos (x) + sin (\frac{3}{2} π) sin (x)

Simplify

cos (\frac{3}{2} π) cos (x) + sin (\frac{3}{2} π) sin (x) : - sin (x)

cos (\frac{3}{2} π) cos (x) + sin (\frac{3}{2} π) sin (x)

cos (\frac{3}{2} π) cos (x) = 0

cos (\frac{3}{2} π) cos (x)

Multiply

\frac{3}{2} π : \frac{3 π}{2}

\frac{3}{2} π

Multiply fractions:

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

= \frac{3 π}{2}

= cos (\frac{3 π}{2}) cos (x)

cos (\frac{3 π}{2}) = 0

cos (\frac{3 π}{2})

Rewrite using trig identities:

cos (π) cos (\frac{π}{2}) - sin (π) sin (\frac{π}{2})

cos (\frac{3 π}{2})

Write

cos (\frac{3 π}{2})

cos (π + \frac{π}{2})

= cos (π + \frac{π}{2})

Use the Angle Sum identity:

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

= cos (π) cos (\frac{π}{2}) - sin (π) sin (\frac{π}{2})

= cos (π) cos (\frac{π}{2}) - sin (π) sin (\frac{π}{2})

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

Use the following trivial identity:

sin (π) = 0

sin (π)

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

= 0

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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

= 1

= (- 1) \cdot 0 - 0 \cdot 1

Simplify

= 0

= 0 \cdot cos (x)

Apply rule

0 \cdot a = 0

= 0

sin (\frac{3}{2} π) sin (x) = - sin (x)

sin (\frac{3}{2} π) sin (x)

Multiply

\frac{3}{2} π : \frac{3 π}{2}

\frac{3}{2} π

Multiply fractions:

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

= \frac{3 π}{2}

= sin (\frac{3 π}{2}) sin (x)

sin (\frac{3 π}{2}) = - 1

sin (\frac{3 π}{2})

Rewrite using trig identities:

sin (π) cos (\frac{π}{2}) + cos (π) sin (\frac{π}{2})

sin (\frac{3 π}{2})

Write

sin (\frac{3 π}{2})

sin (π + \frac{π}{2})

= sin (π + \frac{π}{2})

Use the Angle Sum identity:

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

= sin (π) cos (\frac{π}{2}) + cos (π) sin (\frac{π}{2})

= sin (π) cos (\frac{π}{2}) + cos (π) sin (\frac{π}{2})

Use the following trivial identity:

sin (π) = 0

sin (π)

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

= 0

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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

= 1

= 0 \cdot 0 + (- 1) \cdot 1

Simplify

= - 1

= - 1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= - sin (x)

= 0 - sin (x)

0 - sin (x) = - sin (x)

= - sin (x)

= - sin (x)

- sin (x) + sin (4 x) = 0

Use the Sum to Product identity:

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

= 2 sin (\frac{4 x - x}{2}) cos (\frac{4 x + x}{2})

Simplify

2 sin (\frac{4 x - x}{2}) cos (\frac{4 x + x}{2}) : 2 sin (\frac{3 x}{2}) cos (\frac{5 x}{2})

2 sin (\frac{4 x - x}{2}) cos (\frac{4 x + x}{2})

Add similar elements:

4 x - x = 3 x

= 2 sin (\frac{3 x}{2}) cos (\frac{4 x + x}{2})

Add similar elements:

4 x + x = 5 x

= 2 sin (\frac{3 x}{2}) cos (\frac{5 x}{2})

= 2 sin (\frac{3 x}{2}) cos (\frac{5 x}{2})

2 cos (\frac{5 x}{2}) sin (\frac{3 x}{2}) = 0

Solving each part separately

cos (\frac{5 x}{2}) = 0 or sin (\frac{3 x}{2}) = 0

cos (\frac{5 x}{2}) = 0 : x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

cos (\frac{5 x}{2}) = 0

General solutions for

cos (\frac{5 x}{2}) = 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}

\frac{5 x}{2} = \frac{π}{2} + 2 πn, \frac{5 x}{2} = \frac{3 π}{2} + 2 πn

\frac{5 x}{2} = \frac{π}{2} + 2 πn, \frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{5 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{5} + \frac{4 πn}{5}

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 5 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 5 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

5 x = π + 4 πn

5 x = π + 4 πn

5 x = π + 4 πn

Divide both sides by

5

5 x = π + 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5}

Simplify

x = \frac{π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}

Solve

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{5} + \frac{4 πn}{5}

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 5 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 5 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

Divide both sides by

5

5 x = 3 π + 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{3 π}{5} + \frac{4 πn}{5}

Simplify

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

sin (\frac{3 x}{2}) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{3 x}{2}) = 0

General solutions for

sin (\frac{3 x}{2}) = 0

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

\frac{3 x}{2} = 0 + 2 πn, \frac{3 x}{2} = π + 2 πn

\frac{3 x}{2} = 0 + 2 πn, \frac{3 x}{2} = π + 2 πn

Solve

\frac{3 x}{2} = 0 + 2 πn : x = \frac{4 πn}{3}

\frac{3 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{2} = 2 πn

Multiply both sides by

2

\frac{3 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 2 πn

Simplify

3 x = 4 πn

3 x = 4 πn

Divide both sides by

3

3 x = 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{4 πn}{3}

Simplify

x = \frac{4 πn}{3}

x = \frac{4 πn}{3}

Solve

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

\frac{3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

3 x = 2 π + 4 πn

3 x = 2 π + 4 πn

Divide both sides by

3

3 x = 2 π + 4 πn

Divide both sides by

3

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

Simplify

x = \frac{2 π}{3} + \frac{4 πn}{3}

x = \frac{2 π}{3} + \frac{4 πn}{3}

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

Combine all the solutions

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

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