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Popular >

cos(5x)-cos(3x)=sin(4x)

Solution

cos (5 x) - cos (3 x) = sin (4 x)

Solution

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (5 x) - cos (3 x) = sin (4 x)

Subtract

sin (4 x)

from both sides

cos (5 x) - cos (3 x) - sin (4 x) = 0

Rewrite using trig identities

- cos (3 x) + cos (5 x) - sin (4 x)

Use the Sum to Product identity:

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

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

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

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

\frac{5 x + 3 x}{2} = 4 x

\frac{5 x + 3 x}{2}

Add similar elements:

5 x + 3 x = 8 x

= \frac{8 x}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 x

= 2 sin (4 x) sin (\frac{5 x - 3 x}{2})

\frac{5 x - 3 x}{2} = x

\frac{5 x - 3 x}{2}

Add similar elements:

5 x - 3 x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (4 x) sin (x)

= - sin (4 x) - 2 sin (4 x) sin (x)

- sin (4 x) - 2 sin (4 x) sin (x) = 0

Factor

- sin (4 x) - 2 sin (4 x) sin (x) : - sin (4 x) (2 sin (x) + 1)

- sin (4 x) - 2 sin (4 x) sin (x)

Factor out common term

- sin (4 x)

= - sin (4 x) (1 + 2 sin (x))

- sin (4 x) (2 sin (x) + 1) = 0

Solving each part separately

sin (4 x) = 0 or 2 sin (x) + 1 = 0

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

sin (4 x) = 0

General solutions for

sin (4 x) = 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}

4 x = 0 + 2 πn, 4 x = π + 2 πn

4 x = 0 + 2 πn, 4 x = π + 2 πn

Solve

4 x = 0 + 2 πn : x = \frac{πn}{2}

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

Divide both sides by

4

4 x = 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4}

Simplify

x = \frac{πn}{2}

x = \frac{πn}{2}

Solve

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

4 x = π + 2 πn

Divide both sides by

4

4 x = π + 2 πn

Divide both sides by

4

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

Simplify

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

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

2 sin (x) + 1 = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (x) + 1 = 0

Move

1

to the right side

2 sin (x) + 1 = 0

Subtract

1

from both sides

2 sin (x) + 1 - 1 = 0 - 1

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

Simplify

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{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}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

cos(90-x)+csc(x)= 5/2

cos (9 0^{\circ} - x) + csc (x) = \frac{5}{2}

solvefor x,sin(x)-csc(t)+1=0 solve for

x, sin (x) - csc (t) + 1 = 0

cos^2(x)= 3/(4*5cos^2(x))

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

(cos^2(x))/(1-cos(x))=1+cos(x)

\frac{cos ^{2} ( x )}{1 - cos ( x )} = 1 + cos (x)

sin(x)= 19/50

sin (x) = \frac{19}{50}