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شائع علم المثلثات >

sin(x)+sin(90-x)=sin(90)

الحلّ

sin (x) + sin (9 0^{\circ} - x) = sin (9 0^{\circ})

الحلّ

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

راديان

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

خطوات الحلّ

sin (x) + sin (9 0^{\circ} - x) = sin (9 0^{\circ})

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

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

= 1

= 1

sin (x) + sin (9 0^{\circ} - x) = 1

Rewrite using trig identities

sin (x) + sin (9 0^{\circ} - x)

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

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

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

بسّط:

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

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

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

\frac{x + 9 0 ^{\circ} - x}{2}

x + 9 0^{\circ} - x = 9 0^{\circ}

x + 9 0^{\circ} - x

جمّع التعابير المتشابهة

= x - x + 9 0^{\circ}

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

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

\frac{x - ( 9 0 ^{\circ} - x )}{2}

x - (9 0^{\circ} - x)

وسٌع:

2 x - 9 0^{\circ}

x - (9 0^{\circ} - x)

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

- (9 0^{\circ} - x)

افتح أقواس

= - (9 0^{\circ}) - (- x)

فعّل قوانين سالب-موجب

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

= - 9 0^{\circ} + x

= x - 9 0^{\circ} + x

x - 9 0^{\circ} + x

بسّط:

2 x - 9 0^{\circ}

x - 9 0^{\circ} + x

جمّع التعابير المتشابهة

= x + x - 9 0^{\circ}

x + x = 2 x :

اجمع العناصر المتشابهة

= 2 x - 9 0^{\circ}

= 2 x - 9 0^{\circ}

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

2 x - 9 0^{\circ}

وحّد:

\frac{4 x - 18 0 ^{\circ}}{2}

2 x - 9 0^{\circ}

2 x = \frac{2 x 2}{2}

:حوّل الأعداد لكسور

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

2 \cdot 2 = 4 :

اضرب الأعداد

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

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

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

: استخدم ميزات الكسور التالية

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

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{4 x - 18 0 ^{\circ}}{4}

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

sin (4 5^{\circ})

بسّط:

\frac{2}{2}

sin (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

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

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

= \frac{2}{2}

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

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

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

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

2 cos (\frac{4 x - 18 0 ^{\circ}}{4}) = 1

2

اقسم الطرفين على

2 cos (\frac{4 x - 18 0 ^{\circ}}{4}) = 1

2

اقسم الطرفين على

\frac{2 cos ( \frac{4 x - 18 0 ^{\circ}}{4} )}{2} = \frac{1}{2}

بسّط

\frac{2 cos ( \frac{4 x - 18 0 ^{\circ}}{4} )}{2} = \frac{1}{2}

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

بسّط:

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

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

2 :

إلغ العوامل المشتركة

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

\frac{1}{2}

بسّط:

\frac{2}{2}

\frac{1}{2}

\frac{2}{2}

اضرب بالمرافق

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

a a = a

:فعْل قانون الجذور

22 = 2

= 2

= \frac{2}{2}

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

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

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

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

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 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n, \frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n, \frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n

حلّ:

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

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4}

بسّط:

4 x - 18 0^{\circ}

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4}

\frac{4}{4} = 1 :

اقسم الأعداد

= 4 x - 18 0^{\circ}

4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط:

18 0^{\circ} + 144 0^{\circ} n

4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 4 5^{\circ} = 18 0^{\circ}

4 \cdot 4 5^{\circ}

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

:اضرب كسور

= 18 0^{\circ}

4 :

إلغ العوامل المشتركة

= 18 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

= 18 0^{\circ} + 144 0^{\circ} n

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

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

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

انقل

18 0^{\circ}

إلى الجانب الأيمن

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

للطرفين

18 0^{\circ}

أضف

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

بسّط

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

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

4

اقسم الطرفين على

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

4

اقسم الطرفين على

\frac{4 x}{4} = 9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

بسّط

\frac{4 x}{4} = 9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

\frac{4 x}{4}

بسّط:

x

\frac{4 x}{4}

\frac{4}{4} = 1 :

اقسم الأعداد

= x

9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

بسّط:

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

9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

9 0^{\circ}

اختزل:

9 0^{\circ}

9 0^{\circ}

2 :

إلغ العوامل المشتركة

= 9 0^{\circ}

= 9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

\frac{8}{4} = 2 :

اقسم الأعداد

= 9 0^{\circ} + 36 0^{\circ} n

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

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

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

\frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

حلّ:

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

\frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

4

اضرب الطرفين بـ

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4}

بسّط:

4 x - 18 0^{\circ}

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4}

\frac{4}{4} = 1 :

اقسم الأعداد

= 4 x - 18 0^{\circ}

4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

بسّط:

126 0^{\circ} + 144 0^{\circ} n

4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 31 5^{\circ} = 126 0^{\circ}

4 \cdot 31 5^{\circ}

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

:اضرب كسور

= 126 0^{\circ}

4 :

إلغ العوامل المشتركة

= 126 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

= 126 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

انقل

18 0^{\circ}

إلى الجانب الأيمن

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

للطرفين

18 0^{\circ}

أضف

4 x - 18 0^{\circ} + 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

بسّط

4 x = 144 0^{\circ} + 144 0^{\circ} n

4 x = 144 0^{\circ} + 144 0^{\circ} n

4

اقسم الطرفين على

4 x = 144 0^{\circ} + 144 0^{\circ} n

4

اقسم الطرفين على

\frac{4 x}{4} = 36 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

بسّط

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

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

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

رسم

أمثلة شائعة

sin(2x)=cos(x-60)

sin (2 x) = cos (x - 6 0^{\circ})

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

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

2csc(θ)=sqrt(2)cot(θ)+3csc(θ)

2 csc (θ) = 2 cot (θ) + 3 csc (θ)

arctan(x)=180

arctan (x) = 180

cos(θ)=(2sqrt(2))/3

cos (θ) = \frac{2 2}{3}