What is the general solution for cos(5x-21)=sin(13-3x) ?

The general solution for cos(5x-21)=sin(13-3x) is x=(4pin+16+pi)/4 ,x=-(pi+4pin-68)/(16)

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cos(5x-21)=sin(13-3x)

Solution

cos (5 x - 21) = sin (13 - 3 x)

Solution

x = \frac{4 πn + 16 + π}{4}, x = - \frac{π + 4 πn - 68}{16}

Degrees

x = 274.18311 \dots^{\circ} + 18 0^{\circ} n, x = 232.25706 \dots^{\circ} - 4 5^{\circ} n

Solution steps

cos (5 x - 21) = sin (13 - 3 x)

Rewrite using trig identities

cos (5 x - 21) = sin (13 - 3 x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (5 x - 21) = sin (\frac{π}{2} - (5 x - 21))

cos (5 x - 21) = sin (\frac{π}{2} - (5 x - 21))

Apply trig inverse properties

cos (5 x - 21) = sin (\frac{π}{2} - (5 x - 21))

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

13 - 3 x = \frac{π}{2} - (5 x - 21) + 2 πn, 13 - 3 x = π - (\frac{π}{2} - (5 x - 21)) + 2 πn

13 - 3 x = \frac{π}{2} - (5 x - 21) + 2 πn, 13 - 3 x = π - (\frac{π}{2} - (5 x - 21)) + 2 πn

13 - 3 x = \frac{π}{2} - (5 x - 21) + 2 πn : x = \frac{4 πn + 16 + π}{4}

13 - 3 x = \frac{π}{2} - (5 x - 21) + 2 πn

Expand

\frac{π}{2} - (5 x - 21) + 2 πn : \frac{π}{2} - 5 x + 21 + 2 πn

\frac{π}{2} - (5 x - 21) + 2 πn

- (5 x - 21) : - 5 x + 21

- (5 x - 21)

Distribute parentheses

= - (5 x) - (- 21)

Apply minus-plus rules

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

= - 5 x + 21

= \frac{π}{2} - 5 x + 21 + 2 πn

13 - 3 x = \frac{π}{2} - 5 x + 21 + 2 πn

Move

13

to the right side

13 - 3 x = \frac{π}{2} - 5 x + 21 + 2 πn

Subtract

13

from both sides

13 - 3 x - 13 = \frac{π}{2} - 5 x + 21 + 2 πn - 13

Simplify

13 - 3 x - 13 = \frac{π}{2} - 5 x + 21 + 2 πn - 13

Simplify

13 - 3 x - 13 : - 3 x

13 - 3 x - 13

Add similar elements:

13 - 13 = 0

= - 3 x

Simplify

\frac{π}{2} - 5 x + 21 + 2 πn - 13 : - 5 x + 2 πn + 8 + \frac{π}{2}

\frac{π}{2} - 5 x + 21 + 2 πn - 13

Group like terms

= - 5 x + 2 πn + \frac{π}{2} + 21 - 13

Add/Subtract the numbers:

21 - 13 = 8

= - 5 x + 2 πn + 8 + \frac{π}{2}

- 3 x = - 5 x + 2 πn + 8 + \frac{π}{2}

- 3 x = - 5 x + 2 πn + 8 + \frac{π}{2}

- 3 x = - 5 x + 2 πn + 8 + \frac{π}{2}

Move

5 x

to the left side

- 3 x = - 5 x + 2 πn + 8 + \frac{π}{2}

Add

5 x

to both sides

- 3 x + 5 x = - 5 x + 2 πn + 8 + \frac{π}{2} + 5 x

Simplify

2 x = 2 πn + 8 + \frac{π}{2}

2 x = 2 πn + 8 + \frac{π}{2}

Divide both sides by

2

2 x = 2 πn + 8 + \frac{π}{2}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{8}{2} + \frac{\frac{π}{2}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{8}{2} + \frac{\frac{π}{2}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{8}{2} + \frac{\frac{π}{2}}{2} : \frac{4 πn + 16 + π}{4}

\frac{2 πn}{2} + \frac{8}{2} + \frac{\frac{π}{2}}{2}

Apply rule

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

= \frac{2 πn + 8 + \frac{π}{2}}{2}

Join

2 πn + 8 + \frac{π}{2} : \frac{4 πn + 16 + π}{2}

2 πn + 8 + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 8 = \frac{8 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{8 \cdot 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{2 πn \cdot 2 + 8 \cdot 2 + π}{2}

2 πn \cdot 2 + 8 \cdot 2 + π = 4 πn + 16 + π

2 πn \cdot 2 + 8 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 8 \cdot 2 + π

Multiply the numbers:

8 \cdot 2 = 16

= 4 πn + 16 + π

= \frac{4 πn + 16 + π}{2}

= \frac{\frac{4 πn + 16 + π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 πn + 16 + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 πn + 16 + π}{4}

x = \frac{4 πn + 16 + π}{4}

x = \frac{4 πn + 16 + π}{4}

x = \frac{4 πn + 16 + π}{4}

13 - 3 x = π - (\frac{π}{2} - (5 x - 21)) + 2 πn : x = - \frac{π + 4 πn - 68}{16}

13 - 3 x = π - (\frac{π}{2} - (5 x - 21)) + 2 πn

Expand

π - (\frac{π}{2} - (5 x - 21)) + 2 πn : π - \frac{π}{2} + 5 x - 21 + 2 πn

π - (\frac{π}{2} - (5 x - 21)) + 2 πn

- (5 x - 21) : - 5 x + 21

- (5 x - 21)

Distribute parentheses

= - (5 x) - (- 21)

Apply minus-plus rules

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

= - 5 x + 21

= π - (- 5 x + 21 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 5 x + 21) : - \frac{π}{2} + 5 x - 21

- (\frac{π}{2} - 5 x + 21)

Distribute parentheses

= - (\frac{π}{2}) - (- 5 x) - (21)

Apply minus-plus rules

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

= - \frac{π}{2} + 5 x - 21

= π - \frac{π}{2} + 5 x - 21 + 2 πn

13 - 3 x = π - \frac{π}{2} + 5 x - 21 + 2 πn

Move

13

to the right side

13 - 3 x = π - \frac{π}{2} + 5 x - 21 + 2 πn

Subtract

13

from both sides

13 - 3 x - 13 = π - \frac{π}{2} + 5 x - 21 + 2 πn - 13

Simplify

13 - 3 x - 13 = π - \frac{π}{2} + 5 x - 21 + 2 πn - 13

Simplify

13 - 3 x - 13 : - 3 x

13 - 3 x - 13

Add similar elements:

13 - 13 = 0

= - 3 x

Simplify

π - \frac{π}{2} + 5 x - 21 + 2 πn - 13 : 5 x + 2 πn + π - 34 - \frac{π}{2}

π - \frac{π}{2} + 5 x - 21 + 2 πn - 13

Group like terms

= 5 x + π + 2 πn - \frac{π}{2} - 21 - 13

Subtract the numbers:

- 21 - 13 = - 34

= 5 x + 2 πn + π - 34 - \frac{π}{2}

- 3 x = 5 x + 2 πn + π - 34 - \frac{π}{2}

- 3 x = 5 x + 2 πn + π - 34 - \frac{π}{2}

- 3 x = 5 x + 2 πn + π - 34 - \frac{π}{2}

Move

5 x

to the left side

- 3 x = 5 x + 2 πn + π - 34 - \frac{π}{2}

Subtract

5 x

from both sides

- 3 x - 5 x = 5 x + 2 πn + π - 34 - \frac{π}{2} - 5 x

Simplify

- 8 x = 2 πn + π - 34 - \frac{π}{2}

- 8 x = 2 πn + π - 34 - \frac{π}{2}

Divide both sides by

- 8

- 8 x = 2 πn + π - 34 - \frac{π}{2}

Divide both sides by

- 8

\frac{- 8 x}{- 8} = \frac{2 πn}{- 8} + \frac{π}{- 8} - \frac{34}{- 8} - \frac{\frac{π}{2}}{- 8}

Simplify

\frac{- 8 x}{- 8} = \frac{2 πn}{- 8} + \frac{π}{- 8} - \frac{34}{- 8} - \frac{\frac{π}{2}}{- 8}

Simplify

\frac{- 8 x}{- 8} : x

\frac{- 8 x}{- 8}

Apply the fraction rule:

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

= \frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{2 πn}{- 8} + \frac{π}{- 8} - \frac{34}{- 8} - \frac{\frac{π}{2}}{- 8} : - \frac{π + 4 πn - 68}{16}

\frac{2 πn}{- 8} + \frac{π}{- 8} - \frac{34}{- 8} - \frac{\frac{π}{2}}{- 8}

Apply rule

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

= \frac{2 πn + π - 34 - \frac{π}{2}}{- 8}

Apply the fraction rule:

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

= - \frac{2 πn + π - 34 - \frac{π}{2}}{8}

Join

2 πn + π - 34 - \frac{π}{2} : \frac{π + 4 πn - 68}{2}

2 πn + π - 34 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 34 = \frac{34 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{34 \cdot 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{2 πn \cdot 2 + π 2 - 34 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 34 \cdot 2 - π = π + 4 πn - 68

2 πn \cdot 2 + π 2 - 34 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 34 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 34 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 34 \cdot 2

Multiply the numbers:

34 \cdot 2 = 68

= π + 4 πn - 68

= \frac{π + 4 πn - 68}{2}

= - \frac{\frac{π + 4 πn - 68}{2}}{8}

Simplify

\frac{\frac{π + 4 πn - 68}{2}}{8} : \frac{π + 4 πn - 68}{16}

\frac{\frac{π + 4 πn - 68}{2}}{8}

Apply the fraction rule:

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

= \frac{π + 4 πn - 68}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{π + 4 πn - 68}{16}

= - \frac{π + 4 πn - 68}{16}

x = - \frac{π + 4 πn - 68}{16}

x = - \frac{π + 4 πn - 68}{16}

x = - \frac{π + 4 πn - 68}{16}

x = \frac{4 πn + 16 + π}{4}, x = - \frac{π + 4 πn - 68}{16}

x = \frac{4 πn + 16 + π}{4}, x = - \frac{π + 4 πn - 68}{16}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(5x-21)=sin(13-3x) ?
The general solution for cos(5x-21)=sin(13-3x) is x=(4pin+16+pi)/4 ,x=-(pi+4pin-68)/(16)