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

الحلّ

solve for

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

الحلّ

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}

خطوات الحلّ

sin (α) - sin (β) = cos (β) - cos (α)

من الطرفين

cos (β) - cos (α)

اطرح

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

Rewrite using trig identities

sin (α) - sin (β) - cos (β) + cos (α)

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

أثبت أنّ

sin (α) + cos (α)

أعد الكتابة كـ

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

cos (\frac{π}{4}) = \frac{1}{2} :

استخدم المتطابقة الأساسيّة التالية

sin (\frac{π}{4}) = \frac{1}{2} :

استخدم المتطابقة الأساسيّة التالية

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

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

:فعّل متطابقة الجمع لزوايا

= 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

انقل

cos (β)

إلى الجانب الأيمن

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

للطرفين

cos (β)

أضف

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

بسّط

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

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

انقل

sin (β)

إلى الجانب الأيمن

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

للطرفين

sin (β)

أضف

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

بسّط

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

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

2

اقسم الطرفين على

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

2

اقسم الطرفين على

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

بسّط

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

\frac{2 sin ( α + \frac{π}{4} )}{2}

بسّط:

sin (α + \frac{π}{4})

\frac{2 sin ( α + \frac{π}{4} )}{2}

2 :

إلغ العوامل المشتركة

= sin (α + \frac{π}{4})

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

بسّط:

\frac{2 ( cos ( β ) + sin ( β ) )}{2}

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

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

فعّل القانون

= \frac{cos ( β ) + sin ( β )}{2}

\frac{2}{2}

اضرب بالمرافق

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

22 = 2

22

a a = a

:فعْل قانون الجذور

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}

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

حلول عامّة لـ

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

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

حلّ:

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

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

بسّط:

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}

:فعْل قانون الجذور

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

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

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

:فعّل قانون القوى

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

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

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

اطرح الأعداد

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

:فعْل قانون الجذور

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

انقل

\frac{π}{4}

إلى الجانب الأيمن

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

من الطرفين

\frac{π}{4}

اطرح

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

بسّط

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}

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

حلّ:

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

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

بسّط:

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}

:فعْل قانون الجذور

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

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

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

:فعّل قانون القوى

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

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

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

اطرح الأعداد

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

:فعْل قانون الجذور

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

انقل

\frac{π}{4}

إلى الجانب الأيمن

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

من الطرفين

\frac{π}{4}

اطرح

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

بسّط

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}

أمثلة شائعة

- \frac{6}{7} = tan (t)

sin(2x)=sqrt(3)

sin (2 x) = 3

sin(3x)cos(2x)=0

sin (3 x) cos (2 x) = 0

sin(-2x)=(-1)/2

sin (- 2 x) = \frac{- 1}{2}

2cos(x)sin(x)=1

2 cos (x) sin (x) = 1