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sin(5x-27)=cos(6x+13.6)

Solution

sin (5 x - 27) = cos (6 x + 13.6)

Solution

x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}, x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

Degrees

x = 77.97849 \dots^{\circ} + 32.72727 \dots^{\circ} n, x = - 2416.20864 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (5 x - 27) = cos (6 x + 13.6)

Rewrite using trig identities

sin (5 x - 27) = cos (6 x + 13.6)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x - 27) = sin (\frac{π}{2} - (6 x + 13.6))

sin (5 x - 27) = sin (\frac{π}{2} - (6 x + 13.6))

Apply trig inverse properties

sin (5 x - 27) = sin (\frac{π}{2} - (6 x + 13.6))

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

5 x - 27 = \frac{π}{2} - (6 x + 13.6) + 2 πn, 5 x - 27 = π - (\frac{π}{2} - (6 x + 13.6)) + 2 πn

5 x - 27 = \frac{π}{2} - (6 x + 13.6) + 2 πn, 5 x - 27 = π - (\frac{π}{2} - (6 x + 13.6)) + 2 πn

5 x - 27 = \frac{π}{2} - (6 x + 13.6) + 2 πn : x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}

5 x - 27 = \frac{π}{2} - (6 x + 13.6) + 2 πn

Expand

\frac{π}{2} - (6 x + 13.6) + 2 πn : \frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n

\frac{π}{2} - (6 x + 13.6) + 2 πn

Multiply the numbers:

2 \cdot 3.14159 \dots = 6.28318 \dots

= \frac{π}{2} - (6 x + 13.6) + 6.28318 \dots n

- (6 x + 13.6) : - 6 x - 13.6

- (6 x + 13.6)

Distribute parentheses

= - (6 x) - (13.6)

Apply minus-plus rules

+ (- a) = - a

= - 6 x - 13.6

= \frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n

5 x - 27 = \frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n

Move

27

to the right side

5 x - 27 = \frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n

Add

27

to both sides

5 x - 27 + 27 = \frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n + 27

Simplify

5 x - 27 + 27 = \frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n + 27

Simplify

5 x - 27 + 27 : 5 x

5 x - 27 + 27

Add similar elements:

- 27 + 27 = 0

= 5 x

Simplify

\frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n + 27 : - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2}

\frac{π}{2} - 6 x - 13.6 + 6.28318 \dots n + 27

Group like terms

= - 6 x + 6.28318 \dots n + \frac{π}{2} - 13.6 + 27

Add/Subtract the numbers:

- 13.6 + 27 = 13.4

= - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2}

5 x = - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2}

5 x = - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2}

5 x = - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2}

Move

6 x

to the left side

5 x = - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2}

Add

6 x

to both sides

5 x + 6 x = - 6 x + 6.28318 \dots n + 13.4 + \frac{π}{2} + 6 x

Simplify

11 x = 6.28318 \dots n + 13.4 + \frac{π}{2}

11 x = 6.28318 \dots n + 13.4 + \frac{π}{2}

Divide both sides by

11

11 x = 6.28318 \dots n + 13.4 + \frac{π}{2}

Divide both sides by

11

\frac{11 x}{11} = \frac{6.28318 \dots n}{11} + \frac{13.4}{11} + \frac{\frac{π}{2}}{11}

Simplify

\frac{11 x}{11} = \frac{6.28318 \dots n}{11} + \frac{13.4}{11} + \frac{\frac{π}{2}}{11}

Simplify

\frac{11 x}{11} : x

\frac{11 x}{11}

Divide the numbers:

\frac{11}{11} = 1

= x

Simplify

\frac{6.28318 \dots n}{11} + \frac{13.4}{11} + \frac{\frac{π}{2}}{11} : \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}

\frac{6.28318 \dots n}{11} + \frac{13.4}{11} + \frac{\frac{π}{2}}{11}

Apply rule

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

= \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}

x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}

x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}

x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}

5 x - 27 = π - (\frac{π}{2} - (6 x + 13.6)) + 2 πn : x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

5 x - 27 = π - (\frac{π}{2} - (6 x + 13.6)) + 2 πn

Expand

π - (\frac{π}{2} - (6 x + 13.6)) + 2 πn : π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n

π - (\frac{π}{2} - (6 x + 13.6)) + 2 πn

- (6 x + 13.6) : - 6 x - 13.6

- (6 x + 13.6)

Distribute parentheses

= - (6 x) - (13.6)

Apply minus-plus rules

+ (- a) = - a

= - 6 x - 13.6

= π - (- 6 x + \frac{π}{2} - 13.6) + 2 πn

Multiply the numbers:

2 \cdot 3.14159 \dots = 6.28318 \dots

= π - (- 6 x + \frac{π}{2} - 13.6) + 6.28318 \dots n

- (\frac{π}{2} - 6 x - 13.6) : - \frac{π}{2} + 6 x + 13.6

- (\frac{π}{2} - 6 x - 13.6)

Distribute parentheses

= - (\frac{π}{2}) - (- 6 x) - (- 13.6)

Apply minus-plus rules

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

= - \frac{π}{2} + 6 x + 13.6

= π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n

5 x - 27 = π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n

Move

27

to the right side

5 x - 27 = π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n

Add

27

to both sides

5 x - 27 + 27 = π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n + 27

Simplify

5 x - 27 + 27 = π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n + 27

Simplify

5 x - 27 + 27 : 5 x

5 x - 27 + 27

Add similar elements:

- 27 + 27 = 0

= 5 x

Simplify

π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n + 27 : 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2}

π - \frac{π}{2} + 6 x + 13.6 + 6.28318 \dots n + 27

Group like terms

= 6 x + π + 6.28318 \dots n - \frac{π}{2} + 13.6 + 27

Add the numbers:

13.6 + 27 = 40.6

= 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2}

5 x = 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2}

5 x = 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2}

5 x = 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2}

Move

6 x

to the left side

5 x = 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2}

Subtract

6 x

from both sides

5 x - 6 x = 6 x + 6.28318 \dots n + 40.6 + π - \frac{π}{2} - 6 x

Simplify

- x = 6.28318 \dots n + 40.6 + π - \frac{π}{2}

- x = 6.28318 \dots n + 40.6 + π - \frac{π}{2}

Divide both sides by

- 1

- x = 6.28318 \dots n + 40.6 + π - \frac{π}{2}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{6.28318 \dots n}{- 1} + \frac{40.6}{- 1} + \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{6.28318 \dots n}{- 1} + \frac{40.6}{- 1} + \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{6.28318 \dots n}{- 1} + \frac{40.6}{- 1} + \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} : - 6.28318 \dots n - 40.6 - \frac{π}{2}

\frac{6.28318 \dots n}{- 1} + \frac{40.6}{- 1} + \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1}

Apply rule

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

= \frac{6.28318 \dots n + 40.6 + π - \frac{π}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{6.28318 \dots n + 40.6 + π - \frac{π}{2}}{1}

Join

6.28318 \dots n + 40.6 + π - \frac{π}{2} : 6.28318 \dots n + 40.6 + \frac{π}{2}

6.28318 \dots n + 40.6 + π - \frac{π}{2}

Convert element to fraction:

π = \frac{π 2}{2}

= \frac{π 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{2}

Add similar elements:

2 π - π = π

= \frac{π}{2}

= - \frac{6.28318 \dots n + 40.6 + \frac{π}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{6.28318 \dots n + 40.6 + \frac{π}{2}}{1} = 6.28318 \dots n + 40.6 + \frac{π}{2}

= - (6.28318 \dots n + 40.6 + \frac{π}{2})

Distribute parentheses

= - (6.28318 \dots n) - (40.6) - (\frac{π}{2})

Apply minus-plus rules

+ (- a) = - a

= - 6.28318 \dots n - 40.6 - \frac{π}{2}

x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}, x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

x = \frac{6.28318 \dots n + 13.4 + \frac{π}{2}}{11}, x = - 6.28318 \dots n - 40.6 - \frac{π}{2}

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