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Popular >

solvefor x,cos(2x+5x)=cos(Hx)

Solution

solve for

x, cos (2 x + 5 x) = cos (H x)

Solution

x = \frac{4 πn}{7 + H}, x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}, x = \frac{4 πn}{7 - H}, x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}

Solution steps

cos (2 x + 5 x) = cos (H x)

Subtract

cos (H x)

from both sides

cos (7 x) - cos (x H) = 0

Rewrite using trig identities

cos (7 x) - cos (H x)

Use the Sum to Product identity:

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

= - 2 sin (\frac{7 x + H x}{2}) sin (\frac{7 x - H x}{2})

- 2 sin (\frac{7 x + H x}{2}) sin (\frac{7 x - H x}{2}) = 0

Solving each part separately

sin (\frac{7 x + H x}{2}) = 0 or sin (\frac{7 x - H x}{2}) = 0

sin (\frac{7 x + H x}{2}) = 0 : x = \frac{4 πn}{7 + H}, x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}; H \neq = - 7

sin (\frac{7 x + H x}{2}) = 0

General solutions for

sin (\frac{7 x + H 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{7 x + H x}{2} = 0 + 2 πn, \frac{7 x + H x}{2} = π + 2 πn

\frac{7 x + H x}{2} = 0 + 2 πn, \frac{7 x + H x}{2} = π + 2 πn

Solve

\frac{7 x + H x}{2} = 0 + 2 πn : x = \frac{4 πn}{7 + H}; H \neq = - 7

\frac{7 x + H x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{7 x + H x}{2} = 2 πn

Multiply both sides by

2

\frac{7 x + H x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 7 x + H x )}{2} = 2 \cdot 2 πn

Simplify

7 x + x H = 4 πn

7 x + x H = 4 πn

Factor

7 x + x H : x (7 + H)

7 x + x H

Factor out common term

x

= x (7 + H)

x (7 + H) = 4 πn

Divide both sides by

7 + H; H \neq = - 7

x (7 + H) = 4 πn

Divide both sides by

7 + H; H \neq = - 7

\frac{x ( 7 + H )}{7 + H} = \frac{4 πn}{7 + H}; H \neq = - 7

Simplify

x = \frac{4 πn}{7 + H}; H \neq = - 7

x = \frac{4 πn}{7 + H}; H \neq = - 7

Solve

\frac{7 x + H x}{2} = π + 2 πn : x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}; H \neq = - 7

\frac{7 x + H x}{2} = π + 2 πn

Multiply both sides by

2

\frac{7 x + H x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 7 x + H x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

7 x + x H = 2 π + 4 πn

7 x + x H = 2 π + 4 πn

Factor

7 x + x H : x (7 + H)

7 x + x H

Factor out common term

x

= x (7 + H)

x (7 + H) = 2 π + 4 πn

Divide both sides by

7 + H; H \neq = - 7

x (7 + H) = 2 π + 4 πn

Divide both sides by

7 + H; H \neq = - 7

\frac{x ( 7 + H )}{7 + H} = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}; H \neq = - 7

Simplify

x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}; H \neq = - 7

x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}; H \neq = - 7

x = \frac{4 πn}{7 + H}, x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}; H \neq = - 7

sin (\frac{7 x - H x}{2}) = 0 : x = \frac{4 πn}{7 - H}, x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}; H \neq = 7

sin (\frac{7 x - H x}{2}) = 0

General solutions for

sin (\frac{7 x - H 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{7 x - H x}{2} = 0 + 2 πn, \frac{7 x - H x}{2} = π + 2 πn

\frac{7 x - H x}{2} = 0 + 2 πn, \frac{7 x - H x}{2} = π + 2 πn

Solve

\frac{7 x - H x}{2} = 0 + 2 πn : x = \frac{4 πn}{7 - H}; H \neq = 7

\frac{7 x - H x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{7 x - H x}{2} = 2 πn

Multiply both sides by

2

\frac{7 x - H x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 7 x - H x )}{2} = 2 \cdot 2 πn

Simplify

7 x - x H = 4 πn

7 x - x H = 4 πn

Factor

7 x - x H : x (7 - H)

7 x - x H

Factor out common term

x

= x (7 - H)

x (7 - H) = 4 πn

Divide both sides by

7 - H; H \neq = 7

x (7 - H) = 4 πn

Divide both sides by

7 - H; H \neq = 7

\frac{x ( 7 - H )}{7 - H} = \frac{4 πn}{7 - H}; H \neq = 7

Simplify

x = \frac{4 πn}{7 - H}; H \neq = 7

x = \frac{4 πn}{7 - H}; H \neq = 7

Solve

\frac{7 x - H x}{2} = π + 2 πn : x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}; H \neq = 7

\frac{7 x - H x}{2} = π + 2 πn

Multiply both sides by

2

\frac{7 x - H x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 7 x - H x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

7 x - x H = 2 π + 4 πn

7 x - x H = 2 π + 4 πn

Factor

7 x - x H : x (7 - H)

7 x - x H

Factor out common term

x

= x (7 - H)

x (7 - H) = 2 π + 4 πn

Divide both sides by

7 - H; H \neq = 7

x (7 - H) = 2 π + 4 πn

Divide both sides by

7 - H; H \neq = 7

\frac{x ( 7 - H )}{7 - H} = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}; H \neq = 7

Simplify

x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}; H \neq = 7

x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}; H \neq = 7

x = \frac{4 πn}{7 - H}, x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}; H \neq = 7

Combine all the solutions

x = \frac{4 πn}{7 + H}, x = \frac{2 π}{7 + H} + \frac{4 πn}{7 + H}, x = \frac{4 πn}{7 - H}, x = \frac{2 π}{7 - H} + \frac{4 πn}{7 - H}

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