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شائع علم المثلثات >

cos(x)+2cos(x+240)=1

الحلّ

cos (x) + 2 cos (x + 24 0^{\circ}) = 1

الحلّ

x = 0.61547 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.61547 \dots + 36 0^{\circ} n

راديان

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn

خطوات الحلّ

cos (x) + 2 cos (x + 24 0^{\circ}) = 1

Rewrite using trig identities

cos (x) + 2 cos (x + 24 0^{\circ}) = 1

Rewrite using trig identities

cos (x + 24 0^{\circ})

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

:فعّل متطابقة الجمع لزوايا

= cos (x) cos (24 0^{\circ}) - sin (x) sin (24 0^{\circ})

cos (x) cos (24 0^{\circ}) - sin (x) sin (24 0^{\circ})

بسّط:

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

cos (x) cos (24 0^{\circ}) - sin (x) sin (24 0^{\circ})

cos (x) cos (24 0^{\circ}) = - \frac{1}{2} cos (x)

cos (x) cos (24 0^{\circ})

cos (24 0^{\circ}) = - \frac{1}{2}

cos (24 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

cos (24 0^{\circ})

cos (18 0^{\circ} + 6 0^{\circ})

كـ

cos (24 0^{\circ})

أكتب

= cos (18 0^{\circ} + 6 0^{\circ})

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

:فعّل متطابقة الجمع لزوايا

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

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}

= (- 1)

استخدم المتطابقة الأساسيّة التالية:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

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{1}{2}

استخدم المتطابقة الأساسيّة التالية:

sin (18 0^{\circ}) = 0

sin (18 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}

= 0

استخدم المتطابقة الأساسيّة التالية:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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}

= \frac{3}{2}

= (- 1) \frac{1}{2} - 0 \cdot \frac{3}{2}

بسّط

= - \frac{1}{2}

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

(- a) = - a

:احذف الأقواس

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

= - \frac{1}{2} cos (x) - sin (24 0^{\circ}) sin (x)

sin (24 0^{\circ}) = - \frac{3}{2}

sin (24 0^{\circ})

Rewrite using trig identities:

sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

sin (24 0^{\circ})

sin (18 0^{\circ} + 6 0^{\circ})

كـ

sin (24 0^{\circ})

أكتب

= sin (18 0^{\circ} + 6 0^{\circ})

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

:فعّل متطابقة الجمع لزوايا

= sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

= sin (18 0^{\circ}) cos (6 0^{\circ}) + cos (18 0^{\circ}) sin (6 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (18 0^{\circ}) = 0

sin (18 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}

= 0

استخدم المتطابقة الأساسيّة التالية:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

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{1}{2}

استخدم المتطابقة الأساسيّة التالية:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

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}

= (- 1)

استخدم المتطابقة الأساسيّة التالية:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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}

= \frac{3}{2}

= 0 \cdot \frac{1}{2} + (- 1) \frac{3}{2}

بسّط

= - \frac{3}{2}

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

- (- a) = a

فعّل القانون

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

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

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

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

بسّط:

3 sin (x)

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

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

وسٌع:

- cos (x) + 3 sin (x)

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

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = 2, b = - \frac{1}{2} cos (x), c = \frac{3}{2} sin (x)

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

فعّل قوانين سالب-موجب

+ (- a) = - a

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

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

بسّط:

- cos (x) + 3 sin (x)

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

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

2 \cdot \frac{1}{2} cos (x)

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= cos (x) \cdot 1

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

اضرب

= cos (x)

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

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

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= sin (x) 3

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

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

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

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

اجمع العناصر المتشابهة

= 3 sin (x)

3 sin (x) = 1

3 sin (x) = 1

من الطرفين

1

اطرح

3 sin (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

3 sin (x) - 1 = 0

للطرفين

1

أضف

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

بسّط

3 sin (x) = 1

3 sin (x) = 1

3

اقسم الطرفين على

3 sin (x) = 1

3

اقسم الطرفين على

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

بسّط

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

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

بسّط:

sin (x)

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

3 :

إلغ العوامل المشتركة

= sin (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}

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

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

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

Apply trig inverse properties

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

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

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{3}{3}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{3}) + 36 0^{\circ} n

x = arcsin (\frac{3}{3}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{3}) + 36 0^{\circ} n

أظهر الحلّ بالتمثيل العشريّ

x = 0.61547 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.61547 \dots + 36 0^{\circ} n

رسم

أمثلة شائعة

cos(xpi)=0

cos (x π) = 0

sin^2(2x)= 3/4

sin^{2} (2 x) = \frac{3}{4}

sin(θ)=cos(130)

sin (θ) = cos (13 0^{\circ})

3cot(x)+2=5

3 cot (x) + 2 = 5

sec^2(x)-tan(x)-1=0

sec^{2} (x) - tan (x) - 1 = 0