What is the general solution for sin(3x+87)-cos(7x-17)=0 ?

The general solution for sin(3x+87)-cos(7x-17)=0 is x=(4pin+pi-140}{20},x=-\frac{pi+4pin-208)/8

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sin(3x+87)-cos(7x-17)=0

Solution

sin (3 x + 87) - cos (7 x - 17) = 0

Solution

x = \frac{4 πn + π - 140}{20}, x = - \frac{π + 4 πn - 208}{8}

Degrees

x = - 392.07045 \dots^{\circ} + 3 6^{\circ} n, x = 1467.19026 \dots^{\circ} - 9 0^{\circ} n

Solution steps

sin (3 x + 87) - cos (7 x - 17) = 0

Add

cos (7 x - 17)

to both sides

sin (3 x + 87) = cos (7 x - 17)

Rewrite using trig identities

sin (3 x + 87) = cos (7 x - 17)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (3 x + 87) = sin (\frac{π}{2} - (7 x - 17))

sin (3 x + 87) = sin (\frac{π}{2} - (7 x - 17))

Apply trig inverse properties

sin (3 x + 87) = sin (\frac{π}{2} - (7 x - 17))

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

3 x + 87 = \frac{π}{2} - (7 x - 17) + 2 πn, 3 x + 87 = π - (\frac{π}{2} - (7 x - 17)) + 2 πn

3 x + 87 = \frac{π}{2} - (7 x - 17) + 2 πn, 3 x + 87 = π - (\frac{π}{2} - (7 x - 17)) + 2 πn

3 x + 87 = \frac{π}{2} - (7 x - 17) + 2 πn : x = \frac{4 πn + π - 140}{20}

3 x + 87 = \frac{π}{2} - (7 x - 17) + 2 πn

Expand

\frac{π}{2} - (7 x - 17) + 2 πn : \frac{π}{2} - 7 x + 17 + 2 πn

\frac{π}{2} - (7 x - 17) + 2 πn

- (7 x - 17) : - 7 x + 17

- (7 x - 17)

Distribute parentheses

= - (7 x) - (- 17)

Apply minus-plus rules

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

= - 7 x + 17

= \frac{π}{2} - 7 x + 17 + 2 πn

3 x + 87 = \frac{π}{2} - 7 x + 17 + 2 πn

Move

87

to the right side

3 x + 87 = \frac{π}{2} - 7 x + 17 + 2 πn

Subtract

87

from both sides

3 x + 87 - 87 = \frac{π}{2} - 7 x + 17 + 2 πn - 87

Simplify

3 x + 87 - 87 = \frac{π}{2} - 7 x + 17 + 2 πn - 87

Simplify

3 x + 87 - 87 : 3 x

3 x + 87 - 87

Add similar elements:

87 - 87 = 0

= 3 x

Simplify

\frac{π}{2} - 7 x + 17 + 2 πn - 87 : - 7 x + 2 πn + \frac{π}{2} - 70

\frac{π}{2} - 7 x + 17 + 2 πn - 87

Group like terms

= - 7 x + 2 πn + \frac{π}{2} + 17 - 87

Add/Subtract the numbers:

17 - 87 = - 70

= - 7 x + 2 πn + \frac{π}{2} - 70

3 x = - 7 x + 2 πn + \frac{π}{2} - 70

3 x = - 7 x + 2 πn + \frac{π}{2} - 70

3 x = - 7 x + 2 πn + \frac{π}{2} - 70

Move

7 x

to the left side

3 x = - 7 x + 2 πn + \frac{π}{2} - 70

Add

7 x

to both sides

3 x + 7 x = - 7 x + 2 πn + \frac{π}{2} - 70 + 7 x

Simplify

10 x = 2 πn + \frac{π}{2} - 70

10 x = 2 πn + \frac{π}{2} - 70

Divide both sides by

10

10 x = 2 πn + \frac{π}{2} - 70

Divide both sides by

10

\frac{10 x}{10} = \frac{2 πn}{10} + \frac{\frac{π}{2}}{10} - \frac{70}{10}

Simplify

\frac{10 x}{10} = \frac{2 πn}{10} + \frac{\frac{π}{2}}{10} - \frac{70}{10}

Simplify

\frac{10 x}{10} : x

\frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

\frac{2 πn}{10} + \frac{\frac{π}{2}}{10} - \frac{70}{10} : \frac{4 πn + π - 140}{20}

\frac{2 πn}{10} + \frac{\frac{π}{2}}{10} - \frac{70}{10}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 70}{10}

Join

2 πn + \frac{π}{2} - 70 : \frac{4 πn + π - 140}{2}

2 πn + \frac{π}{2} - 70

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 70 = \frac{70 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{70 \cdot 2}{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 - 70 \cdot 2}{2}

2 πn \cdot 2 + π - 70 \cdot 2 = 4 πn + π - 140

2 πn \cdot 2 + π - 70 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 70 \cdot 2

Multiply the numbers:

70 \cdot 2 = 140

= 4 πn + π - 140

= \frac{4 πn + π - 140}{2}

= \frac{\frac{4 πn + π - 140}{2}}{10}

Apply the fraction rule:

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

= \frac{4 πn + π - 140}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{4 πn + π - 140}{20}

x = \frac{4 πn + π - 140}{20}

x = \frac{4 πn + π - 140}{20}

x = \frac{4 πn + π - 140}{20}

3 x + 87 = π - (\frac{π}{2} - (7 x - 17)) + 2 πn : x = - \frac{π + 4 πn - 208}{8}

3 x + 87 = π - (\frac{π}{2} - (7 x - 17)) + 2 πn

Expand

π - (\frac{π}{2} - (7 x - 17)) + 2 πn : π - \frac{π}{2} + 7 x - 17 + 2 πn

π - (\frac{π}{2} - (7 x - 17)) + 2 πn

- (7 x - 17) : - 7 x + 17

- (7 x - 17)

Distribute parentheses

= - (7 x) - (- 17)

Apply minus-plus rules

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

= - 7 x + 17

= π - (- 7 x + 17 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 7 x + 17) : - \frac{π}{2} + 7 x - 17

- (\frac{π}{2} - 7 x + 17)

Distribute parentheses

= - (\frac{π}{2}) - (- 7 x) - (17)

Apply minus-plus rules

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

= - \frac{π}{2} + 7 x - 17

= π - \frac{π}{2} + 7 x - 17 + 2 πn

3 x + 87 = π - \frac{π}{2} + 7 x - 17 + 2 πn

Move

87

to the right side

3 x + 87 = π - \frac{π}{2} + 7 x - 17 + 2 πn

Subtract

87

from both sides

3 x + 87 - 87 = π - \frac{π}{2} + 7 x - 17 + 2 πn - 87

Simplify

3 x + 87 - 87 = π - \frac{π}{2} + 7 x - 17 + 2 πn - 87

Simplify

3 x + 87 - 87 : 3 x

3 x + 87 - 87

Add similar elements:

87 - 87 = 0

= 3 x

Simplify

π - \frac{π}{2} + 7 x - 17 + 2 πn - 87 : 7 x + 2 πn + π - 104 - \frac{π}{2}

π - \frac{π}{2} + 7 x - 17 + 2 πn - 87

Group like terms

= 7 x + π + 2 πn - \frac{π}{2} - 17 - 87

Subtract the numbers:

- 17 - 87 = - 104

= 7 x + 2 πn + π - 104 - \frac{π}{2}

3 x = 7 x + 2 πn + π - 104 - \frac{π}{2}

3 x = 7 x + 2 πn + π - 104 - \frac{π}{2}

3 x = 7 x + 2 πn + π - 104 - \frac{π}{2}

Move

7 x

to the left side

3 x = 7 x + 2 πn + π - 104 - \frac{π}{2}

Subtract

7 x

from both sides

3 x - 7 x = 7 x + 2 πn + π - 104 - \frac{π}{2} - 7 x

Simplify

- 4 x = 2 πn + π - 104 - \frac{π}{2}

- 4 x = 2 πn + π - 104 - \frac{π}{2}

Divide both sides by

- 4

- 4 x = 2 πn + π - 104 - \frac{π}{2}

Divide both sides by

- 4

\frac{- 4 x}{- 4} = \frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{104}{- 4} - \frac{\frac{π}{2}}{- 4}

Simplify

\frac{- 4 x}{- 4} = \frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{104}{- 4} - \frac{\frac{π}{2}}{- 4}

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{104}{- 4} - \frac{\frac{π}{2}}{- 4} : - \frac{π + 4 πn - 208}{8}

\frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{104}{- 4} - \frac{\frac{π}{2}}{- 4}

Apply rule

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

= \frac{2 πn + π - 104 - \frac{π}{2}}{- 4}

Apply the fraction rule:

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

= - \frac{2 πn + π - 104 - \frac{π}{2}}{4}

Join

2 πn + π - 104 - \frac{π}{2} : \frac{π + 4 πn - 208}{2}

2 πn + π - 104 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 104 = \frac{104 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{104 \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 - 104 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 104 \cdot 2 - π = π + 4 πn - 208

2 πn \cdot 2 + π 2 - 104 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 104 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 104 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 104 \cdot 2

Multiply the numbers:

104 \cdot 2 = 208

= π + 4 πn - 208

= \frac{π + 4 πn - 208}{2}

= - \frac{\frac{π + 4 πn - 208}{2}}{4}

Simplify

\frac{\frac{π + 4 πn - 208}{2}}{4} : \frac{π + 4 πn - 208}{8}

\frac{\frac{π + 4 πn - 208}{2}}{4}

Apply the fraction rule:

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

= \frac{π + 4 πn - 208}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 4 πn - 208}{8}

= - \frac{π + 4 πn - 208}{8}

x = - \frac{π + 4 πn - 208}{8}

x = - \frac{π + 4 πn - 208}{8}

x = - \frac{π + 4 πn - 208}{8}

x = \frac{4 πn + π - 140}{20}, x = - \frac{π + 4 πn - 208}{8}

x = \frac{4 πn + π - 140}{20}, x = - \frac{π + 4 πn - 208}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x+87)-cos(7x-17)=0 ?
The general solution for sin(3x+87)-cos(7x-17)=0 is x=(4pin+pi-140}{20},x=-\frac{pi+4pin-208)/8