What is the general solution for cos(2x+5)=sin(2x) ?

The general solution for cos(2x+5)=sin(2x) is x=pin+pi/8-5/4 ,x=(5pi)/8-5/4+pin

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

Solution

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

Solution

x = πn + \frac{π}{8} - \frac{5}{4}, x = \frac{5 π}{8} - \frac{5}{4} + πn

Degrees

x = - 49.11972 \dots^{\circ} + 18 0^{\circ} n, x = 40.88027 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

sin (2 x)

from both sides

cos (2 x + 5) - sin (2 x) = 0

Rewrite using trig identities

cos (5 + 2 x) - sin (2 x)

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

= cos (5 + 2 x) - cos (\frac{π}{2} - 2 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{5 + 2 x + \frac{π}{2} - 2 x}{2}) sin (\frac{5 + 2 x - ( \frac{π}{2} - 2 x )}{2})

Simplify

- 2 sin (\frac{5 + 2 x + \frac{π}{2} - 2 x}{2}) sin (\frac{5 + 2 x - ( \frac{π}{2} - 2 x )}{2}) : - 2 sin (\frac{π + 10}{4}) sin (\frac{8 x - π + 10}{4})

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

\frac{5 + 2 x + \frac{π}{2} - 2 x}{2} = \frac{π + 10}{4}

\frac{5 + 2 x + \frac{π}{2} - 2 x}{2}

5 + 2 x + \frac{π}{2} - 2 x = \frac{π}{2} + 5

5 + 2 x + \frac{π}{2} - 2 x

Group like terms

= 2 x - 2 x + \frac{π}{2} + 5

Add similar elements:

2 x - 2 x = 0

= \frac{π}{2} + 5

= \frac{\frac{π}{2} + 5}{2}

Join

\frac{π}{2} + 5 : \frac{π + 10}{2}

\frac{π}{2} + 5

Convert element to fraction:

5 = \frac{5 \cdot 2}{2}

= \frac{π}{2} + \frac{5 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 5 \cdot 2}{2}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{π + 10}{2}

= \frac{\frac{π + 10}{2}}{2}

Apply the fraction rule:

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

= \frac{π + 10}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 10}{4}

= - 2 sin (\frac{π + 10}{4}) sin (\frac{2 x - ( - 2 x + \frac{π}{2} ) + 5}{2})

\frac{5 + 2 x - ( \frac{π}{2} - 2 x )}{2} = \frac{8 x - π + 10}{4}

\frac{5 + 2 x - ( \frac{π}{2} - 2 x )}{2}

Expand

5 + 2 x - (\frac{π}{2} - 2 x) : 4 x - \frac{π}{2} + 5

5 + 2 x - (\frac{π}{2} - 2 x)

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

- (\frac{π}{2} - 2 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 2 x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + 2 x

= 5 + 2 x - \frac{π}{2} + 2 x

Simplify

5 + 2 x - \frac{π}{2} + 2 x : 4 x - \frac{π}{2} + 5

5 + 2 x - \frac{π}{2} + 2 x

Group like terms

= 2 x + 2 x - \frac{π}{2} + 5

Add similar elements:

2 x + 2 x = 4 x

= 4 x - \frac{π}{2} + 5

= 4 x - \frac{π}{2} + 5

= \frac{4 x - \frac{π}{2} + 5}{2}

Join

4 x - \frac{π}{2} + 5 : \frac{8 x - π + 10}{2}

4 x - \frac{π}{2} + 5

Convert element to fraction:

4 x = \frac{4 x 2}{2}, 5 = \frac{5 \cdot 2}{2}

= \frac{4 x \cdot 2}{2} - \frac{π}{2} + \frac{5 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{4 x \cdot 2 - π + 5 \cdot 2}{2}

4 x \cdot 2 - π + 5 \cdot 2 = 8 x - π + 10

4 x \cdot 2 - π + 5 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= 8 x - π + 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 8 x - π + 10

= \frac{8 x - π + 10}{2}

= \frac{\frac{8 x - π + 10}{2}}{2}

Apply the fraction rule:

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

= \frac{8 x - π + 10}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8 x - π + 10}{4}

= - 2 sin (\frac{π + 10}{4}) sin (\frac{8 x + 10 - π}{4})

= - 2 sin (\frac{π + 10}{4}) sin (\frac{8 x - π + 10}{4})

- 2 sin (\frac{π + 10}{4}) sin (\frac{8 x - π + 10}{4}) = 0

Divide both sides by

- 2 sin (\frac{π + 10}{4})

- 2 sin (\frac{π + 10}{4}) sin (\frac{8 x - π + 10}{4}) = 0

Divide both sides by

- 2 sin (\frac{π + 10}{4})

\frac{- 2 sin ( \frac{π + 10}{4} ) sin ( \frac{8 x - π + 10}{4} )}{- 2 sin ( \frac{π + 10}{4} )} = \frac{0}{- 2 sin ( \frac{π + 10}{4} )}

Simplify

sin (\frac{8 x - π + 10}{4}) = 0

sin (\frac{8 x - π + 10}{4}) = 0

General solutions for

sin (\frac{8 x - π + 10}{4}) = 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{8 x - π + 10}{4} = 0 + 2 πn, \frac{8 x - π + 10}{4} = π + 2 πn

\frac{8 x - π + 10}{4} = 0 + 2 πn, \frac{8 x - π + 10}{4} = π + 2 πn

Solve

\frac{8 x - π + 10}{4} = 0 + 2 πn : x = πn + \frac{π}{8} - \frac{5}{4}

\frac{8 x - π + 10}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{8 x - π + 10}{4} = 2 πn

Multiply both sides by

4

\frac{8 x - π + 10}{4} = 2 πn

Multiply both sides by

4

\frac{4 ( 8 x - π + 10 )}{4} = 4 \cdot 2 πn

Simplify

8 x - π + 10 = 8 πn

8 x - π + 10 = 8 πn

Move

π

to the right side

8 x - π + 10 = 8 πn

Add

π

to both sides

8 x - π + 10 + π = 8 πn + π

Simplify

8 x + 10 = 8 πn + π

8 x + 10 = 8 πn + π

Move

10

to the right side

8 x + 10 = 8 πn + π

Subtract

10

from both sides

8 x + 10 - 10 = 8 πn + π - 10

Simplify

8 x = 8 πn + π - 10

8 x = 8 πn + π - 10

Divide both sides by

8

8 x = 8 πn + π - 10

Divide both sides by

8

\frac{8 x}{8} = \frac{8 πn}{8} + \frac{π}{8} - \frac{10}{8}

Simplify

\frac{8 x}{8} = \frac{8 πn}{8} + \frac{π}{8} - \frac{10}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{8 πn}{8} + \frac{π}{8} - \frac{10}{8} : πn + \frac{π}{8} - \frac{5}{4}

\frac{8 πn}{8} + \frac{π}{8} - \frac{10}{8}

Divide the numbers:

\frac{8}{8} = 1

= πn + \frac{π}{8} - \frac{10}{8}

Cancel

\frac{10}{8} : \frac{5}{4}

\frac{10}{8}

Cancel the common factor:

2

= \frac{5}{4}

= πn + \frac{π}{8} - \frac{5}{4}

x = πn + \frac{π}{8} - \frac{5}{4}

x = πn + \frac{π}{8} - \frac{5}{4}

x = πn + \frac{π}{8} - \frac{5}{4}

Solve

\frac{8 x - π + 10}{4} = π + 2 πn : x = \frac{5 π}{8} - \frac{5}{4} + πn

\frac{8 x - π + 10}{4} = π + 2 πn

Multiply both sides by

4

\frac{8 x - π + 10}{4} = π + 2 πn

Multiply both sides by

4

\frac{4 ( 8 x - π + 10 )}{4} = 4 π + 4 \cdot 2 πn

Simplify

8 x - π + 10 = 4 π + 8 πn

8 x - π + 10 = 4 π + 8 πn

Move

π

to the right side

8 x - π + 10 = 4 π + 8 πn

Add

π

to both sides

8 x - π + 10 + π = 4 π + 8 πn + π

Simplify

8 x + 10 = 5 π + 8 πn

8 x + 10 = 5 π + 8 πn

Move

10

to the right side

8 x + 10 = 5 π + 8 πn

Subtract

10

from both sides

8 x + 10 - 10 = 5 π + 8 πn - 10

Simplify

8 x = 5 π + 8 πn - 10

8 x = 5 π + 8 πn - 10

Divide both sides by

8

8 x = 5 π + 8 πn - 10

Divide both sides by

8

\frac{8 x}{8} = \frac{5 π}{8} + \frac{8 πn}{8} - \frac{10}{8}

Simplify

\frac{8 x}{8} = \frac{5 π}{8} + \frac{8 πn}{8} - \frac{10}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{5 π}{8} + \frac{8 πn}{8} - \frac{10}{8} : \frac{5 π}{8} - \frac{5}{4} + πn

\frac{5 π}{8} + \frac{8 πn}{8} - \frac{10}{8}

Group like terms

= \frac{5 π}{8} - \frac{10}{8} + \frac{8 πn}{8}

Cancel

\frac{10}{8} : \frac{5}{4}

\frac{10}{8}

Cancel the common factor:

2

= \frac{5}{4}

= \frac{5 π}{8} - \frac{5}{4} + \frac{8 πn}{8}

Divide the numbers:

\frac{8}{8} = 1

= \frac{5 π}{8} - \frac{5}{4} + πn

x = \frac{5 π}{8} - \frac{5}{4} + πn

x = \frac{5 π}{8} - \frac{5}{4} + πn

x = \frac{5 π}{8} - \frac{5}{4} + πn

x = πn + \frac{π}{8} - \frac{5}{4}, x = \frac{5 π}{8} - \frac{5}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x+5)=sin(2x) ?
The general solution for cos(2x+5)=sin(2x) is x=pin+pi/8-5/4 ,x=(5pi)/8-5/4+pin