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شائع علم المثلثات >

cos(x+1/6 pi)cos(x-1/6 pi)=cos(2x)

الحلّ

cos (x + \frac{1}{6} π) cos (x - \frac{1}{6} π) = cos (2 x)

الحلّ

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

درجات

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

خطوات الحلّ

cos (x + \frac{1}{6} π) cos (x - \frac{1}{6} π) = cos (2 x)

Rewrite using trig identities

cos (x + \frac{1}{6} π) cos (x - \frac{1}{6} π) = cos (2 x)

Rewrite using trig identities

cos (x + \frac{1}{6} π)

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

:فعّل متطابقة الجمع لزوايا

= cos (x) cos (\frac{1}{6} π) - sin (x) sin (\frac{1}{6} π)

cos (x) cos (\frac{1}{6} π) - sin (x) sin (\frac{1}{6} π)

بسّط:

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

cos (x) cos (\frac{1}{6} π) - sin (x) sin (\frac{1}{6} π)

cos (x) cos (\frac{1}{6} π) = \frac{3}{2} cos (x)

cos (x) cos (\frac{1}{6} π)

\frac{1}{6} π

اضرب بـ:

\frac{π}{6}

\frac{1}{6} π

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

:اضرب كسور

= \frac{1 π}{6}

1 π = π :

اضرب

= \frac{π}{6}

= cos (\frac{π}{6}) cos (x)

cos (\frac{π}{6})

بسّط:

\frac{3}{2}

cos (\frac{π}{6})

استخدم المتطابقة الأساسيّة التالية:

cos (\frac{π}{6}) = \frac{3}{2}

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}

= \frac{3}{2}

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

sin (x) sin (\frac{1}{6} π) = \frac{1}{2} sin (x)

sin (x) sin (\frac{1}{6} π)

\frac{1}{6} π

اضرب بـ:

\frac{π}{6}

\frac{1}{6} π

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

:اضرب كسور

= \frac{1 π}{6}

1 π = π :

اضرب

= \frac{π}{6}

= sin (\frac{π}{6}) sin (x)

sin (\frac{π}{6})

بسّط:

\frac{1}{2}

sin (\frac{π}{6})

استخدم المتطابقة الأساسيّة التالية:

sin (\frac{π}{6}) = \frac{1}{2}

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{1}{2}

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

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

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

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

:فعّل متطابقة الطرح لزوايا

= cos (x) cos (\frac{1}{6} π) + sin (x) sin (\frac{1}{6} π)

cos (x) cos (\frac{1}{6} π) + sin (x) sin (\frac{1}{6} π)

بسّط:

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

cos (x) cos (\frac{1}{6} π) + sin (x) sin (\frac{1}{6} π)

cos (x) cos (\frac{1}{6} π) = \frac{3}{2} cos (x)

cos (x) cos (\frac{1}{6} π)

\frac{1}{6} π

اضرب بـ:

\frac{π}{6}

\frac{1}{6} π

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

:اضرب كسور

= \frac{1 π}{6}

1 π = π :

اضرب

= \frac{π}{6}

= cos (\frac{π}{6}) cos (x)

cos (\frac{π}{6})

بسّط:

\frac{3}{2}

cos (\frac{π}{6})

استخدم المتطابقة الأساسيّة التالية:

cos (\frac{π}{6}) = \frac{3}{2}

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}

= \frac{3}{2}

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

sin (x) sin (\frac{1}{6} π) = \frac{1}{2} sin (x)

sin (x) sin (\frac{1}{6} π)

\frac{1}{6} π

اضرب بـ:

\frac{π}{6}

\frac{1}{6} π

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

:اضرب كسور

= \frac{1 π}{6}

1 π = π :

اضرب

= \frac{π}{6}

= sin (\frac{π}{6}) sin (x)

sin (\frac{π}{6})

بسّط:

\frac{1}{2}

sin (\frac{π}{6})

استخدم المتطابقة الأساسيّة التالية:

sin (\frac{π}{6}) = \frac{1}{2}

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{1}{2}

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

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

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

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

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

من الطرفين

cos (2 x)

اطرح

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

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

بسّط:

\frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) ) - 4 cos ( 2 x )}{4}

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

(\frac{3}{2} cos (x) - \frac{1}{2} sin (x)) (\frac{3}{2} cos (x) + \frac{1}{2} sin (x)) = \frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) )}{4}

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

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

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

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

:اضرب كسور

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

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

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

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

:اضرب كسور

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

1 \cdot sin (x) = sin (x) :

اضرب

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

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

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

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

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

:اضرب كسور

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

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

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

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

:اضرب كسور

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

1 \cdot sin (x) = sin (x) :

اضرب

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

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

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

بسّط:

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

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

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

فعّل القانون

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

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

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

وحّد الكسور:

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

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

فعّل القانون

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

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

(a) = a

:احذف الأقواس

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

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

:اضرب كسور

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

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) )}{4}

= \frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) )}{4} - cos (2 x)

cos (2 x) = \frac{cos ( 2 x ) 4}{4}

:حوّل الأعداد لكسور

= \frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) )}{4} - \frac{cos ( 2 x ) \cdot 4}{4}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) ) - cos ( 2 x ) \cdot 4}{4}

\frac{( 3 cos ( x ) - sin ( x ) ) ( 3 cos ( x ) + sin ( x ) ) - 4 cos ( 2 x )}{4} = 0

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

(3 cos (x) - sin (x)) (3 cos (x) + sin (x)) - 4 cos (2 x) = 0

Rewrite using trig identities

(- sin (x) + cos (x) 3) (sin (x) + cos (x) 3) - 4 cos (2 x)

cos (2 x) = cos^{2} (x) - sin^{2} (x)

:فعّل متطابقة الزاوية المضاعفة

= (- sin (x) + 3 cos (x)) (sin (x) + 3 cos (x)) - 4 (cos^{2} (x) - sin^{2} (x))

(- sin (x) + 3 cos (x)) (sin (x) + 3 cos (x)) - 4 (cos^{2} (x) - sin^{2} (x))

بسّط:

- cos^{2} (x) + 3 sin^{2} (x)

(- sin (x) + 3 cos (x)) (sin (x) + 3 cos (x)) - 4 (cos^{2} (x) - sin^{2} (x))

(- sin (x) + 3 cos (x)) (sin (x) + 3 cos (x))

وسٌع:

3 cos^{2} (x) - sin^{2} (x)

(- sin (x) + 3 cos (x)) (sin (x) + 3 cos (x))

(a - b) (a + b) = a^{2} - b^{2}

فعّل قانون فرق المربّعات

a = 3 cos (x), b = sin (x)

= (3 cos (x))^{2} - sin^{2} (x)

(3 cos (x))^{2} = 3 cos^{2} (x)

(3 cos (x))^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= (3)^{2} cos^{2} (x)

(3)^{2} : 3

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (3^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 3

= 3 cos^{2} (x)

= 3 cos^{2} (x) - sin^{2} (x)

= 3 cos^{2} (x) - sin^{2} (x) - 4 (cos^{2} (x) - sin^{2} (x))

- 4 (cos^{2} (x) - sin^{2} (x))

وسٌع:

- 4 cos^{2} (x) + 4 sin^{2} (x)

- 4 (cos^{2} (x) - sin^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 4, b = cos^{2} (x), c = sin^{2} (x)

= - 4 cos^{2} (x) - (- 4) sin^{2} (x)

فعّل قوانين سالب-موجب

- (- a) = a

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= 3 cos^{2} (x) - sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

3 cos^{2} (x) - sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

بسّط:

- cos^{2} (x) + 3 sin^{2} (x)

3 cos^{2} (x) - sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

3 cos^{2} (x) - 4 cos^{2} (x) = - cos^{2} (x) :

اجمع العناصر المتشابهة

= - cos^{2} (x) - sin^{2} (x) + 4 sin^{2} (x)

- sin^{2} (x) + 4 sin^{2} (x) = 3 sin^{2} (x) :

اجمع العناصر المتشابهة

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

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

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

- cos^{2} (x) + 3 sin^{2} (x) = 0

- cos^{2} (x) + 3 sin^{2} (x)

حلل إلى عوامل:

(3 sin (x) + cos (x)) (3 sin (x) - cos (x))

- cos^{2} (x) + 3 sin^{2} (x)

(3 sin (x))^{2} - cos^{2} (x)

كـ

3 sin^{2} (x) - cos^{2} (x)

اكتب مجددًا

3 sin^{2} (x) - cos^{2} (x)

a = (a)^{2}

:فعْل قانون الجذور

3 = (3)^{2}

= (3)^{2} sin^{2} (x) - cos^{2} (x)

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

(3)^{2} sin^{2} (x) = (3 sin (x))^{2}

= (3 sin (x))^{2} - cos^{2} (x)

= (3 sin (x))^{2} - cos^{2} (x)

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

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

= (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

(3 sin (x) + cos (x)) (3 sin (x) - cos (x)) = 0

حلّ كل جزء على حدة

3 sin (x) + cos (x) = 0 or 3 sin (x) - cos (x) = 0

3 sin (x) + cos (x) = 0 : x = \frac{5 π}{6} + πn

3 sin (x) + cos (x) = 0

Rewrite using trig identities

3 sin (x) + cos (x) = 0

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

اقسم الطرفين على

\frac{3 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{3 sin ( x )}{cos ( x )} + 1 = 0

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

:Use the basic trigonometric identity

3 tan (x) + 1 = 0

3 tan (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

3 tan (x) + 1 = 0

من الطرفين

1

اطرح

3 tan (x) + 1 - 1 = 0 - 1

بسّط

3 tan (x) = - 1

3 tan (x) = - 1

3

اقسم الطرفين على

3 tan (x) = - 1

3

اقسم الطرفين على

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

بسّط

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

\frac{3 tan ( x )}{3}

بسّط:

tan (x)

\frac{3 tan ( x )}{3}

3 :

إلغ العوامل المشتركة

= tan (x)

\frac{- 1}{3}

بسّط:

- \frac{3}{3}

\frac{- 1}{3}

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

: استخدم ميزات الكسور التالية

= - \frac{1}{3}

- \frac{1}{3}

حوّل لصيغة عدد كسريّ:

- \frac{3}{3}

- \frac{1}{3}

\frac{3}{3}

اضرب بالمرافق

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3} :

حلول عامّة لـ

tan (x)

periodicity table with

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

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

3 sin (x) - cos (x) = 0 : x = \frac{π}{6} + πn

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

Rewrite using trig identities

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

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

اقسم الطرفين على

\frac{3 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

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

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

:Use the basic trigonometric identity

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

3 tan (x) - 1 = 0

للطرفين

1

أضف

3 tan (x) - 1 + 1 = 0 + 1

بسّط

3 tan (x) = 1

3 tan (x) = 1

3

اقسم الطرفين على

3 tan (x) = 1

3

اقسم الطرفين على

\frac{3 tan ( x )}{3} = \frac{1}{3}

بسّط

\frac{3 tan ( x )}{3} = \frac{1}{3}

\frac{3 tan ( x )}{3}

بسّط:

tan (x)

\frac{3 tan ( x )}{3}

3 :

إلغ العوامل المشتركة

= tan (x)

\frac{1}{3}

بسّط:

\frac{3}{3}

\frac{1}{3}

\frac{3}{3}

اضرب بالمرافق

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3} :

حلول عامّة لـ

tan (x)

periodicity table with

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

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

وحّد الحلول

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

رسم

أمثلة شائعة

sin(x)=0.731

sin (x) = 0.731

sin(x)=0.766

sin (x) = 0.766

sin(x)=0.755

sin (x) = 0.755

3tan^2(x)-5tan(x)+2=0

3 tan^{2} (x) - 5 tan (x) + 2 = 0

solvefor θ,ρ=(4sin(φ)cos(θ))

so l v e f or θ, ρ = (4 sin (φ) cos (θ))