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شائع علم المثلثات >

tan(x/2)=sin(x)

الحلّ

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

الحلّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 4 πn, x = 2 π + 4 πn

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 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}) = sin (x)

من الطرفين

sin (x)

اطرح

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة تحويل الضرب لجمع

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

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

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

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

اضرب

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

(a) = a

:احذف الأقواس

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

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

x + \frac{x}{2}

وحّد:

\frac{3 x}{2}

x + \frac{x}{2}

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

x - \frac{x}{2}

وحّد:

\frac{x}{2}

x - \frac{x}{2}

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

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

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

:حوّل الأعداد لكسور

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

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

وسٌع:

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

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

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

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

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

افتح أقواس

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

فعّل قوانين سالب-موجب

+ (- a) = - a

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

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

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

اجمع العناصر المتشابهة

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

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

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

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

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

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

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

بسّط:

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

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

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

وحّد الكسور:

2 x

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

فعّل القانون

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

x + 3 x = 4 x :

اجمع العناصر المتشابهة

= \frac{4 x}{2}

\frac{4}{2} = 2 :

اقسم الأعداد

= 2 x

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

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

وحّد الكسور:

- x

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

فعّل القانون

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

x - 3 x = - 2 x :

اجمع العناصر المتشابهة

= \frac{- 2 x}{2}

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

: استخدم ميزات الكسور التالية

= - \frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= - x

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

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

: استخدم ميزات الكسور التالية

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

\frac{2}{2} = 1 :

اقسم الأعداد

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

sin (- x) = - sin (x)

:Use the negative angle identity

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

(- a) = - a

:احذف الأقواس

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

\frac{2}{2} = 1 :

اقسم الأعداد

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

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

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

حلّ كل جزء على حدة

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0 :

حلول عامّة لـ

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 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 = \frac{3 π}{2} + 2 πn, x = 4 πn, x = 2 π + 4 πn

رسم

أمثلة شائعة

7sin(B)+sqrt(23)=0

7 sin (B) + 23 = 0

cos^2(3x)=1

cos^{2} (3 x) = 1

tan(pi/4-x)+cot(pi/4-x)=4

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

7sin(θ)-1=5sin(θ)

7 sin (θ) - 1 = 5 sin (θ)

csc(θ)=-7/3

csc (θ) = - \frac{7}{3}