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شائع علم المثلثات >

solvefor a,sin(a+b/2)=cos(c/2)

الحلّ

solve for

a, sin (a + \frac{b}{2}) = cos (\frac{c}{2})

الحلّ

a = \frac{π - c - b}{2} + 2 πn, a = π + 2 πn + \frac{- π + c - b}{2}

خطوات الحلّ

sin (a + \frac{b}{2}) = cos (\frac{c}{2})

Rewrite using trig identities

cos (\frac{c}{2})

cos (x) = sin (\frac{π}{2} - x)

:استخدم المتطابقة التالية

sin (\frac{π}{2} - \frac{c}{2})

sin (a + \frac{b}{2}) = sin (\frac{π}{2} - \frac{c}{2})

Apply trig inverse properties

sin (a + \frac{b}{2}) = sin (\frac{π}{2} - \frac{c}{2})

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

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn, a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn, a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn : a = \frac{π - c - b}{2} + 2 πn

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn

انقل

\frac{b}{2}

إلى الجانب الأيمن

a + \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn

من الطرفين

\frac{b}{2}

اطرح

a + \frac{b}{2} - \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

بسّط

a + \frac{b}{2} - \frac{b}{2} = \frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

a + \frac{b}{2} - \frac{b}{2}

بسّط:

a

a + \frac{b}{2} - \frac{b}{2}

\frac{b}{2} - \frac{b}{2} = 0 :

اجمع العناصر المتشابهة

= a

\frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

بسّط:

\frac{π - c - b}{2} + 2 πn

\frac{π}{2} - \frac{c}{2} + 2 πn - \frac{b}{2}

\frac{π}{2} - \frac{c}{2} - \frac{b}{2}

وحّد الكسور:

\frac{π - c - b}{2}

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

فعّل القانون

= \frac{π - c - b}{2}

= \frac{π - b - c}{2} + 2 πn

= \frac{π - c - b}{2} + 2 πn

a = \frac{π - c - b}{2} + 2 πn

a = \frac{π - c - b}{2} + 2 πn

a = \frac{π - c - b}{2} + 2 πn

a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn : a = π + 2 πn + \frac{- π + c - b}{2}

a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

انقل

\frac{b}{2}

إلى الجانب الأيمن

a + \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn

من الطرفين

\frac{b}{2}

اطرح

a + \frac{b}{2} - \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

بسّط

a + \frac{b}{2} - \frac{b}{2} = π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

a + \frac{b}{2} - \frac{b}{2}

بسّط:

a

a + \frac{b}{2} - \frac{b}{2}

\frac{b}{2} - \frac{b}{2} = 0 :

اجمع العناصر المتشابهة

= a

π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

بسّط:

π + 2 πn + \frac{- π + c - b}{2}

π - (\frac{π}{2} - \frac{c}{2}) + 2 πn - \frac{b}{2}

\frac{π}{2} - \frac{c}{2}

وحّد الكسور:

\frac{π - c}{2}

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

فعّل القانون

= \frac{π - c}{2}

= π - \frac{- c + π}{2} + 2 πn - \frac{b}{2}

(a) = a

:احذف الأقواس

= π - \frac{π - c}{2} + 2 πn - \frac{b}{2}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- ( π - c ) - b}{2}

- (π - c) : - π + c

- (π - c)

افتح أقواس

= - π - (- c)

فعّل قوانين سالب-موجب

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

= - π + c

= π + 2 πn + \frac{c - b - π}{2}

a = π + 2 πn + \frac{- π + c - b}{2}

a = π + 2 πn + \frac{- π + c - b}{2}

a = π + 2 πn + \frac{- π + c - b}{2}

a = \frac{π - c - b}{2} + 2 πn, a = π + 2 πn + \frac{- π + c - b}{2}

رسم

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