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شائع >

solvefor θ,cos(θ)+cos(90-θ)=0

الحلّ

solve for

θ, cos (θ) + cos (9 0^{\circ} - θ) = 0

الحلّ

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

راديان

θ = \frac{3 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

خطوات الحلّ

cos (θ) + cos (9 0^{\circ} - θ) = 0

Rewrite using trig identities

cos (θ) + cos (9 0^{\circ} - θ)

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

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

= 2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

بسّط:

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

\frac{θ + 9 0 ^{\circ} - θ}{2} = 4 5^{\circ}

\frac{θ + 9 0 ^{\circ} - θ}{2}

θ + 9 0^{\circ} - θ = 9 0^{\circ}

θ + 9 0^{\circ} - θ

جمّع التعابير المتشابهة

= θ - θ + 9 0^{\circ}

θ - θ = 0 :

اجمع العناصر المتشابهة

= 9 0^{\circ}

= \frac{9 0 ^{\circ}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{18 0 ^{\circ}}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= 4 5^{\circ}

= 2 cos (4 5^{\circ}) cos (\frac{θ - ( - θ + 9 0 ^{\circ} )}{2})

\frac{θ - ( 9 0 ^{\circ} - θ )}{2} = \frac{4 θ - 18 0 ^{\circ}}{4}

\frac{θ - ( 9 0 ^{\circ} - θ )}{2}

θ - (9 0^{\circ} - θ)

وسٌع:

2 θ - 9 0^{\circ}

θ - (9 0^{\circ} - θ)

- (9 0^{\circ} - θ) : - 9 0^{\circ} + θ

- (9 0^{\circ} - θ)

افتح أقواس

= - (9 0^{\circ}) - (- θ)

فعّل قوانين سالب-موجب

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

= - 9 0^{\circ} + θ

= θ - 9 0^{\circ} + θ

θ - 9 0^{\circ} + θ

بسّط:

2 θ - 9 0^{\circ}

θ - 9 0^{\circ} + θ

جمّع التعابير المتشابهة

= θ + θ - 9 0^{\circ}

θ + θ = 2 θ :

اجمع العناصر المتشابهة

= 2 θ - 9 0^{\circ}

= 2 θ - 9 0^{\circ}

= \frac{2 θ - 9 0 ^{\circ}}{2}

2 θ - 9 0^{\circ}

وحّد:

\frac{4 θ - 18 0 ^{\circ}}{2}

2 θ - 9 0^{\circ}

2 θ = \frac{2 θ 2}{2}

:حوّل الأعداد لكسور

= \frac{2 θ \cdot 2}{2} - 9 0^{\circ}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{2 θ \cdot 2 - 18 0 ^{\circ}}{2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{4 θ - 18 0 ^{\circ}}{2}

= \frac{\frac{4 θ - 18 0 ^{\circ}}{2}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{4 θ - 18 0 ^{\circ}}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{4 θ - 18 0 ^{\circ}}{4}

= 2 cos (4 5^{\circ}) cos (\frac{4 θ - 18 0 ^{\circ}}{4})

cos (4 5^{\circ})

بسّط:

\frac{2}{2}

cos (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= 2 \cdot \frac{2}{2} cos (\frac{4 θ - 18 0 ^{\circ}}{4})

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

:اضرب كسور

= \frac{2 \cdot 2 cos ( \frac{4 θ - 18 0 ^{\circ}}{4} )}{2}

2 :

إلغ العوامل المشتركة

= 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

= 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

2

اقسم الطرفين على

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

2

اقسم الطرفين على

\frac{2 cos ( \frac{4 θ - 18 0 ^{\circ}}{4} )}{2} = \frac{0}{2}

بسّط

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0 :

حلول عامّة لـ

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n, \frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n, \frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

حلّ:

θ = 13 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

بسّط:

4 θ - 18 0^{\circ}

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

\frac{4}{4} = 1 :

اقسم الأعداد

= 4 θ - 18 0^{\circ}

4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط:

36 0^{\circ} + 144 0^{\circ} n

4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 9 0^{\circ} = 36 0^{\circ}

4 \cdot 9 0^{\circ}

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

:اضرب كسور

= 36 0^{\circ}

\frac{4}{2} = 2 :

اقسم الأعداد

= 36 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

4 \cdot 2 = 8 :

اضرب الأعداد

= 144 0^{\circ} n

= 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

انقل

18 0^{\circ}

إلى الجانب الأيمن

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

للطرفين

18 0^{\circ}

أضف

4 θ - 18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

بسّط

4 θ = 54 0^{\circ} + 144 0^{\circ} n

4 θ = 54 0^{\circ} + 144 0^{\circ} n

4

اقسم الطرفين على

4 θ = 54 0^{\circ} + 144 0^{\circ} n

4

اقسم الطرفين على

\frac{4 θ}{4} = 13 5^{\circ} + \frac{144 0 ^{\circ} n}{4}

بسّط

θ = 13 5^{\circ} + 36 0^{\circ} n

θ = 13 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

حلّ:

θ = 31 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

بسّط:

4 θ - 18 0^{\circ}

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

\frac{4}{4} = 1 :

اقسم الأعداد

= 4 θ - 18 0^{\circ}

4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط:

108 0^{\circ} + 144 0^{\circ} n

4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 27 0^{\circ} = 108 0^{\circ}

4 \cdot 27 0^{\circ}

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

:اضرب كسور

= 108 0^{\circ}

3 \cdot 4 = 12 :

اضرب الأعداد

= 108 0^{\circ}

\frac{12}{2} = 6 :

اقسم الأعداد

= 108 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

4 \cdot 2 = 8 :

اضرب الأعداد

= 144 0^{\circ} n

= 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

انقل

18 0^{\circ}

إلى الجانب الأيمن

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

للطرفين

18 0^{\circ}

أضف

4 θ - 18 0^{\circ} + 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

بسّط

4 θ = 126 0^{\circ} + 144 0^{\circ} n

4 θ = 126 0^{\circ} + 144 0^{\circ} n

4

اقسم الطرفين على

4 θ = 126 0^{\circ} + 144 0^{\circ} n

4

اقسم الطرفين على

\frac{4 θ}{4} = 31 5^{\circ} + \frac{144 0 ^{\circ} n}{4}

بسّط

θ = 31 5^{\circ} + 36 0^{\circ} n

θ = 31 5^{\circ} + 36 0^{\circ} n

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

رسم

أمثلة شائعة

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