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شائع علم المثلثات >

cos(45+x)-cos(45-x)=sqrt(2)cos(x)

الحلّ

cos (4 5^{\circ} + x) - cos (4 5^{\circ} - x) = 2 cos (x)

الحلّ

x = 13 5^{\circ} + 18 0^{\circ} n

راديان

x = \frac{3 π}{4} + πn

خطوات الحلّ

cos (4 5^{\circ} + x) - cos (4 5^{\circ} - x) = 2 cos (x)

Rewrite using trig identities

cos (4 5^{\circ} + x) - cos (4 5^{\circ} - x) = 2 cos (x)

Rewrite using trig identities

cos (4 5^{\circ} - x)

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

:فعّل متطابقة الطرح لزوايا

= cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x)

cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x)

بسّط:

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

cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x)

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

cos (4 5^{\circ}) cos (x)

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}

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

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

:اضرب كسور

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

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

sin (4 5^{\circ}) sin (x)

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}

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

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

:اضرب كسور

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

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

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

فعّل القانون

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

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

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

:فعّل متطابقة الجمع لزوايا

= cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

بسّط:

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

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

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

cos (4 5^{\circ}) cos (x)

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}

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

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

:اضرب كسور

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

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

sin (4 5^{\circ}) sin (x)

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}

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

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

:اضرب كسور

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

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

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

فعّل القانون

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

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

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

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

بسّط:

- 2 sin (x)

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

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

فعّل القانون

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

2 cos (x) - 2 sin (x) - (2 cos (x) + 2 sin (x))

وسٌع:

- 22 sin (x)

2 cos (x) - 2 sin (x) - (2 cos (x) + 2 sin (x))

- (2 cos (x) + 2 sin (x)) : - 2 cos (x) - 2 sin (x)

- (2 cos (x) + 2 sin (x))

افتح أقواس

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

فعّل قوانين سالب-موجب

+ (- a) = - a

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

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

2 cos (x) - 2 sin (x) - 2 cos (x) - 2 sin (x)

بسّط:

- 22 sin (x)

2 cos (x) - 2 sin (x) - 2 cos (x) - 2 sin (x)

2 cos (x) - 2 cos (x) = 0 :

اجمع العناصر المتشابهة

= - 2 sin (x) - 2 sin (x)

- 2 sin (x) - 2 sin (x) = - 22 sin (x) :

اجمع العناصر المتشابهة

= - 22 sin (x)

= - 22 sin (x)

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

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

: استخدم ميزات الكسور التالية

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

\frac{2}{2} = 1 :

اقسم الأعداد

= - 2 sin (x)

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

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

من الطرفين

2 cos (x)

اطرح

- 2 sin (x) - 2 cos (x) = 0

Rewrite using trig identities

- 2 sin (x) - 2 cos (x) = 0

cos (x) \neq = 0, cos (x)

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

- 2 tan (x) - 2 = 0

- 2 tan (x) - 2 = 0

انقل

2

إلى الجانب الأيمن

- 2 tan (x) - 2 = 0

للطرفين

2

أضف

- 2 tan (x) - 2 + 2 = 0 + 2

بسّط

- 2 tan (x) = 2

- 2 tan (x) = 2

- 2

اقسم الطرفين على

- 2 tan (x) = 2

- 2

اقسم الطرفين على

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

بسّط

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1 :

حلول عامّة لـ

tan (x)

periodicity table with

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 13 5^{\circ} + 18 0^{\circ} n

x = 13 5^{\circ} + 18 0^{\circ} n

رسم

أمثلة شائعة

sin(b)=cos(2b-30)

sin (b) = cos (2 b - 30)

2sin(x)cos(x)-4cos(x)-sin(x)+2=0

2 sin (x) cos (x) - 4 cos (x) - sin (x) + 2 = 0

(sin(pi/2))/5 =(sin(x))/4

\frac{sin ( \frac{π}{2} )}{5} = \frac{sin ( x )}{4}

cos(3x-1)= 1/2

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

sin(b)=(sqrt(2))/2

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