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Popular >

solvefor α,sin(α)-sin(β)=cos(β)-cos(α)

Solution

solve for

α, sin (α) - sin (β) = cos (β) - cos (α)

Solution

α = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}, α = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

Solution steps

sin (α) - sin (β) = cos (β) - cos (α)

Subtract

cos (β) - cos (α)

from both sides

sin (α) - sin (β) - cos (β) + cos (α) = 0

Rewrite using trig identities

sin (α) - sin (β) - cos (β) + cos (α)

Prove the identity:

sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

sin (α) + cos (α)

Rewrite as

= 2 (\frac{1}{2} sin (α) + \frac{1}{2} cos (α))

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (α) + sin (\frac{π}{4}) cos (α))

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (α + \frac{π}{4})

= - cos (β) - sin (β) + 2 sin (α + \frac{π}{4})

- cos (β) - sin (β) + 2 sin (α + \frac{π}{4}) = 0

Move

cos (β)

to the right side

- cos (β) - sin (β) + 2 sin (α + \frac{π}{4}) = 0

Add

cos (β)

to both sides

- cos (β) - sin (β) + 2 sin (α + \frac{π}{4}) + cos (β) = 0 + cos (β)

Simplify

- sin (β) + 2 sin (α + \frac{π}{4}) = cos (β)

- sin (β) + 2 sin (α + \frac{π}{4}) = cos (β)

Move

sin (β)

to the right side

- sin (β) + 2 sin (α + \frac{π}{4}) = cos (β)

Add

sin (β)

to both sides

- sin (β) + 2 sin (α + \frac{π}{4}) + sin (β) = cos (β) + sin (β)

Simplify

2 sin (α + \frac{π}{4}) = cos (β) + sin (β)

2 sin (α + \frac{π}{4}) = cos (β) + sin (β)

Divide both sides by

2

2 sin (α + \frac{π}{4}) = cos (β) + sin (β)

Divide both sides by

2

\frac{2 sin ( α + \frac{π}{4} )}{2} = \frac{cos ( β )}{2} + \frac{sin ( β )}{2}

Simplify

\frac{2 sin ( α + \frac{π}{4} )}{2} = \frac{cos ( β )}{2} + \frac{sin ( β )}{2}

Simplify

\frac{2 sin ( α + \frac{π}{4} )}{2} : sin (α + \frac{π}{4})

\frac{2 sin ( α + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (α + \frac{π}{4})

Simplify

\frac{cos ( β )}{2} + \frac{sin ( β )}{2} : \frac{2 ( cos ( β ) + sin ( β ) )}{2}

\frac{cos ( β )}{2} + \frac{sin ( β )}{2}

Apply rule

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

= \frac{cos ( β ) + sin ( β )}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( cos ( β ) + sin ( β ) ) 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2 ( cos ( β ) + sin ( β ) )}{2}

sin (α + \frac{π}{4}) = \frac{2 ( cos ( β ) + sin ( β ) )}{2}

sin (α + \frac{π}{4}) = \frac{2 ( cos ( β ) + sin ( β ) )}{2}

sin (α + \frac{π}{4}) = \frac{2 ( cos ( β ) + sin ( β ) )}{2}

Apply trig inverse properties

sin (α + \frac{π}{4}) = \frac{2 ( cos ( β ) + sin ( β ) )}{2}

General solutions for

sin (α + \frac{π}{4}) = \frac{2 ( cos ( β ) + sin ( β ) )}{2}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

α + \frac{π}{4} = arcsin (\frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn, α + \frac{π}{4} = π + arcsin (- \frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn

α + \frac{π}{4} = arcsin (\frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn, α + \frac{π}{4} = π + arcsin (- \frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn

Solve

α + \frac{π}{4} = arcsin (\frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn : α = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

α + \frac{π}{4} = arcsin (\frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn

Simplify

arcsin (\frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn : arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn

arcsin (\frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn

\frac{2 ( cos ( β ) + sin ( β ) )}{2} = \frac{cos ( β ) + sin ( β )}{2}

\frac{2 ( cos ( β ) + sin ( β ) )}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} ( cos ( β ) + sin ( β ) )}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{cos ( β ) + sin ( β )}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{cos ( β ) + sin ( β )}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{cos ( β ) + sin ( β )}{2}

= arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn

α + \frac{π}{4} = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn

Move

\frac{π}{4}

to the right side

α + \frac{π}{4} = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn

Subtract

\frac{π}{4}

from both sides

α + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

Simplify

α = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

α = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

Solve

α + \frac{π}{4} = π + arcsin (- \frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn : α = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

α + \frac{π}{4} = π + arcsin (- \frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn

Simplify

π + arcsin (- \frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn : π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn

π + arcsin (- \frac{2 ( cos ( β ) + sin ( β ) )}{2}) + 2 πn

\frac{2 ( cos ( β ) + sin ( β ) )}{2} = \frac{cos ( β ) + sin ( β )}{2}

\frac{2 ( cos ( β ) + sin ( β ) )}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} ( cos ( β ) + sin ( β ) )}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{cos ( β ) + sin ( β )}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{cos ( β ) + sin ( β )}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{cos ( β ) + sin ( β )}{2}

= π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn

α + \frac{π}{4} = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn

Move

\frac{π}{4}

to the right side

α + \frac{π}{4} = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn

Subtract

\frac{π}{4}

from both sides

α + \frac{π}{4} - \frac{π}{4} = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

Simplify

α = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

α = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

α = arcsin (\frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}, α = π + arcsin (- \frac{cos ( β ) + sin ( β )}{2}) + 2 πn - \frac{π}{4}

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