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شائع علم المثلثات >

tan(x/2)+cos(x)=1

الحلّ

tan (\frac{x}{2}) + cos (x) = 1

الحلّ

x = \frac{π}{2} + 2 πn, x = 4 πn, x = 2 π + 4 πn

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

خطوات الحلّ

tan (\frac{x}{2}) + cos (x) = 1

من الطرفين

1

اطرح

tan (\frac{x}{2}) + cos (x) - 1 = 0

sin, cos :

عبّر بواسطة

- 1 + cos (x) + tan (\frac{x}{2})

tan (x) = \frac{sin ( x )}{cos ( x )}

:Use the basic trigonometric identity

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

- 1 + cos (x) + \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

بسّط:

\frac{- cos ( \frac{x}{2} ) + cos ( x ) cos ( \frac{x}{2} ) + sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

- 1 + cos (x) + \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

1 = \frac{1 cos ( \frac{x}{2} )}{cos ( \frac{x}{2} )}, cos (x) = \frac{cos ( x ) cos ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot cos (\frac{x}{2}) = cos (\frac{x}{2}) :

اضرب

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

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

\frac{- cos ( \frac{x}{2} ) + sin ( \frac{x}{2} ) + cos ( \frac{x}{2} ) cos ( x )}{cos ( \frac{x}{2} )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- cos (\frac{x}{2}) + sin (\frac{x}{2}) + cos (\frac{x}{2}) cos (x) = 0

Rewrite using trig identities

- cos (\frac{x}{2}) + sin (\frac{x}{2}) + cos (\frac{x}{2}) cos (x)

cos (s) cos (t) = \frac{1}{2} (cos (s - t) + cos (s + t))

:فعّل متطابقة تحويل الضرب لجمع

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

- cos (\frac{x}{2}) + sin (\frac{x}{2}) + \frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

بسّط:

\frac{- cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) + 2 sin ( \frac{x}{2} )}{2}

- cos (\frac{x}{2}) + sin (\frac{x}{2}) + \frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

\frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)) = \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

\frac{1}{2} (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

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

:اضرب كسور

= \frac{1 \cdot ( cos ( \frac{x}{2} - x ) + cos ( \frac{x}{2} + x ) )}{2}

1 \cdot (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)) = cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)

1 \cdot (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

1 \cdot (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)) = (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)) :

اضرب

= (cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x))

(a) = a

:احذف الأقواس

= cos (\frac{x}{2} - x) + cos (\frac{x}{2} + x)

= \frac{cos ( \frac{x}{2} - x ) + cos ( \frac{x}{2} + x )}{2}

\frac{x}{2} - x

وحّد:

- \frac{x}{2}

\frac{x}{2} - x

x = \frac{x 2}{2}

:حوّل الأعداد لكسور

= \frac{x}{2} - \frac{x \cdot 2}{2}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{x - x \cdot 2}{2}

x - 2 x = - x :

اجمع العناصر المتشابهة

= \frac{- x}{2}

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

: استخدم ميزات الكسور التالية

= - \frac{x}{2}

= \frac{cos ( - \frac{x}{2} ) + cos ( \frac{x}{2} + x )}{2}

\frac{x}{2} + x

وحّد:

\frac{3 x}{2}

\frac{x}{2} + x

x = \frac{x 2}{2}

:حوّل الأعداد لكسور

= \frac{x}{2} + \frac{x \cdot 2}{2}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{x + x \cdot 2}{2}

x + 2 x = 3 x :

اجمع العناصر المتشابهة

= \frac{3 x}{2}

= \frac{cos ( - \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

cos (- \frac{x}{2}) + cos (\frac{3 x}{2})

بسّط:

cos (\frac{x}{2}) + cos (\frac{3 x}{2})

cos (- \frac{x}{2}) + cos (\frac{3 x}{2})

cos (- x) = cos (x)

:Use the negative angle identity

= cos (\frac{x}{2}) + cos (\frac{3 x}{2})

= \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

= - cos (\frac{x}{2}) + sin (\frac{x}{2}) + \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2}

cos (\frac{x}{2}) = \frac{cos ( \frac{x}{2} ) 2}{2}, sin (\frac{x}{2}) = \frac{sin ( \frac{x}{2} ) 2}{2}

:حوّل الأعداد لكسور

= \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} )}{2} - \frac{cos ( \frac{x}{2} ) \cdot 2}{2} + \frac{sin ( \frac{x}{2} ) \cdot 2}{2}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) - cos ( \frac{x}{2} ) \cdot 2 + sin ( \frac{x}{2} ) \cdot 2}{2}

cos (\frac{x}{2}) + cos (\frac{3 x}{2}) - cos (\frac{x}{2}) \cdot 2 + sin (\frac{x}{2}) \cdot 2 = - cos (\frac{x}{2}) + cos (\frac{3 x}{2}) + 2 sin (\frac{x}{2})

cos (\frac{x}{2}) + cos (\frac{3 x}{2}) - cos (\frac{x}{2}) \cdot 2 + sin (\frac{x}{2}) \cdot 2

جمّع التعابير المتشابهة

= cos (\frac{x}{2}) + cos (\frac{3 x}{2}) - 2 cos (\frac{x}{2}) + 2 sin (\frac{x}{2})

cos (\frac{x}{2}) - 2 cos (\frac{x}{2}) = - cos (\frac{x}{2}) :

اجمع العناصر المتشابهة

= - cos (\frac{x}{2}) + cos (\frac{3 x}{2}) + 2 sin (\frac{x}{2})

= \frac{- cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) + 2 sin ( \frac{x}{2} )}{2}

= \frac{- cos ( \frac{x}{2} ) + cos ( \frac{3 x}{2} ) + 2 sin ( \frac{x}{2} )}{2}

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

= \frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{\frac{3 x}{2} - \frac{x}{2}}{2} )}{2}

\frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{\frac{3 x}{2} - \frac{x}{2}}{2} )}{2}

بسّط:

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

\frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{\frac{3 x}{2} - \frac{x}{2}}{2} )}{2}

\frac{3 x}{2} - \frac{x}{2}

وحّد الكسور:

x

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

فعّل القانون

= \frac{3 x - x}{2}

3 x - x = 2 x :

اجمع العناصر المتشابهة

= \frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

= \frac{2 sin ( \frac{x}{2} ) - 2 sin ( \frac{\frac{3 x}{2} + \frac{x}{2}}{2} ) sin ( \frac{x}{2} )}{2}

\frac{3 x}{2} + \frac{x}{2}

وحّد الكسور:

2 x

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

فعّل القانون

= \frac{3 x + x}{2}

3 x + x = 4 x :

اجمع العناصر المتشابهة

= \frac{4 x}{2}

\frac{4}{2} = 2 :

اقسم الأعداد

= 2 x

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

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

حلل إلى عوامل:

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

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

أعد الكتابة كـ

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

2 sin (\frac{x}{2})

قم باخراج العامل المشترك

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

بسّط

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

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

\frac{2}{2} = 1 :

اقسم الأعداد

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

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

(1 - sin (x)) sin (\frac{x}{2}) = 0

حلّ كل جزء على حدة

1 - sin (x) = 0 or sin (\frac{x}{2}) = 0

1 - sin (x) = 0 : x = \frac{π}{2} + 2 πn

1 - sin (x) = 0

انقل

1

إلى الجانب الأيمن

1 - sin (x) = 0

من الطرفين

1

اطرح

1 - sin (x) - 1 = 0 - 1

بسّط

- sin (x) = - 1

- sin (x) = - 1

- 1

اقسم الطرفين على

- sin (x) = - 1

- 1

اقسم الطرفين على

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

بسّط

sin (x) = 1

sin (x) = 1

sin (x) = 1 :

حلول عامّة لـ

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) = 0

sin (\frac{x}{2}) = 0 :

حلول عامّة لـ

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn

حلّ:

x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

2

اضرب الطرفين بـ

\frac{x}{2} = 2 πn

2

اضرب الطرفين بـ

\frac{2 x}{2} = 2 \cdot 2 πn

بسّط

x = 4 πn

x = 4 πn

\frac{x}{2} = π + 2 πn

حلّ:

x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

2

اضرب الطرفين بـ

\frac{x}{2} = π + 2 πn

2

اضرب الطرفين بـ

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

بسّط

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = 4 πn, x = 2 π + 4 πn

رسم

أمثلة شائعة

6cos^2(x)-7cos(x)-5=0