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شائع علم المثلثات >

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

الحلّ

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

الحلّ

x = 0.54408 \dots + 2 πn - \frac{π}{4}, x = π - 0.54408 \dots + 2 πn - \frac{π}{4}

درجات

x = - 13.82604 \dots^{\circ} + 36 0^{\circ} n, x = 103.82604 \dots^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

3 - sin (x)

اطرح

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

Rewrite using trig identities

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

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

sin (x) + cos (x)

أعد الكتابة كـ

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

cos (\frac{π}{4}) = \frac{1}{2} :

استخدم المتطابقة الأساسيّة التالية

sin (\frac{π}{4}) = \frac{1}{2} :

استخدم المتطابقة الأساسيّة التالية

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

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

:فعّل متطابقة الجمع لزوايا

= 2 sin (x + \frac{π}{4})

= 1 - 3 + 2 sin (x + \frac{π}{4})

1 - 3 + 2 sin (x + \frac{π}{4}) = 0

انقل

1

إلى الجانب الأيمن

1 - 3 + 2 sin (x + \frac{π}{4}) = 0

من الطرفين

1

اطرح

1 - 3 + 2 sin (x + \frac{π}{4}) - 1 = 0 - 1

بسّط

- 3 + 2 sin (x + \frac{π}{4}) = - 1

- 3 + 2 sin (x + \frac{π}{4}) = - 1

انقل

3

إلى الجانب الأيمن

- 3 + 2 sin (x + \frac{π}{4}) = - 1

للطرفين

3

أضف

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

بسّط

2 sin (x + \frac{π}{4}) = - 1 + 3

2 sin (x + \frac{π}{4}) = - 1 + 3

2

اقسم الطرفين على

2 sin (x + \frac{π}{4}) = - 1 + 3

2

اقسم الطرفين على

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

بسّط

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

\frac{2 sin ( x + \frac{π}{4} )}{2}

بسّط:

sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

2 :

إلغ العوامل المشتركة

= sin (x + \frac{π}{4})

- \frac{1}{2} + \frac{3}{2}

بسّط:

\frac{2 ( - 1 + 3 )}{2}

- \frac{1}{2} + \frac{3}{2}

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

فعّل القانون

= \frac{- 1 + 3}{2}

\frac{2}{2}

اضرب بالمرافق

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

22 = 2

22

a a = a

:فعْل قانون الجذور

22 = 2

= 2

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

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

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

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

Apply trig inverse properties

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

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

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

حلّ:

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

بسّط:

arcsin (\frac{- 1 + 3}{2}) + 2 πn

arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

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

\frac{2 ( - 1 + 3 )}{2}

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

:فعْل قانون الجذور

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

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

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

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

:فعْل قانون الجذور

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

= \frac{- 1 + 3}{2}

= arcsin (\frac{3 - 1}{2}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn

انقل

\frac{π}{4}

إلى الجانب الأيمن

x + \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn

من الطرفين

\frac{π}{4}

اطرح

x + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

بسّط

x + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} - \frac{π}{4}

بسّط:

x

x + \frac{π}{4} - \frac{π}{4}

\frac{π}{4} - \frac{π}{4} = 0 :

اجمع العناصر المتشابهة

= x

arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

بسّط:

arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

= arcsin (\frac{2 ( 3 - 1 )}{2}) + 2 πn - \frac{π}{4}

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

\frac{2 ( - 1 + 3 )}{2}

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

:فعْل قانون الجذور

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

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

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

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

:فعْل قانون الجذور

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

= \frac{- 1 + 3}{2}

= arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

Could not simplify further

= arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

حلّ:

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

بسّط:

π - arcsin (\frac{- 1 + 3}{2}) + 2 πn

π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

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

\frac{2 ( - 1 + 3 )}{2}

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

:فعْل قانون الجذور

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

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

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

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

:فعْل قانون الجذور

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

= \frac{- 1 + 3}{2}

= π - arcsin (\frac{3 - 1}{2}) + 2 πn

x + \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn

انقل

\frac{π}{4}

إلى الجانب الأيمن

x + \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn

من الطرفين

\frac{π}{4}

اطرح

x + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

بسّط

x + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} - \frac{π}{4}

بسّط:

x

x + \frac{π}{4} - \frac{π}{4}

\frac{π}{4} - \frac{π}{4} = 0 :

اجمع العناصر المتشابهة

= x

π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

بسّط:

π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

= π - arcsin (\frac{2 ( 3 - 1 )}{2}) + 2 πn - \frac{π}{4}

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

\frac{2 ( - 1 + 3 )}{2}

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

:فعْل قانون الجذور

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

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

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

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

:فعْل قانون الجذور

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

= \frac{- 1 + 3}{2}

= π - arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

Could not simplify further

= π - arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}, x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

أظهر الحلّ بالتمثيل العشريّ

x = 0.54408 \dots + 2 πn - \frac{π}{4}, x = π - 0.54408 \dots + 2 πn - \frac{π}{4}

رسم

أمثلة شائعة

3=7cos(pi/3 t)

3 = 7 cos (\frac{π}{3} t)

cos^2(2x)=cos(4x)

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

solvefor θ,tan(2θ)=-1

so l v e f or θ, tan (2 θ) = - 1

3cos(z+1.2)=2

3 cos (z + 1.2) = 2

1/2 =sqrt(3/2)*cos(x)

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