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شائع علم المثلثات >

sec^2(x)-1= 1/(cot(x))

الحلّ

sec^{2} (x) - 1 = \frac{1}{cot ( x )}

الحلّ

x = \frac{π}{4} + πn

درجات

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

خطوات الحلّ

sec^{2} (x) - 1 = \frac{1}{cot ( x )}

ربّع الطرفين

(sec^{2} (x) - 1)^{2} = (\frac{1}{cot ( x )})^{2}

من الطرفين

(\frac{1}{cot ( x )})^{2}

اطرح

(sec^{2} (x) - 1)^{2} - \frac{1}{cot ^{2} ( x )} = 0

(sec^{2} (x) - 1)^{2} - \frac{1}{cot ^{2} ( x )}

بسّط:

\frac{cot ^{2} ( x ) ( sec ^{2} ( x ) - 1 ) ^{2} - 1}{cot ^{2} ( x )}

(sec^{2} (x) - 1)^{2} - \frac{1}{cot ^{2} ( x )}

(sec^{2} (x) - 1)^{2} = \frac{( sec ^{2} ( x ) - 1 ) ^{2} cot ^{2} ( x )}{cot ^{2} ( x )}

:حوّل الأعداد لكسور

= \frac{( sec ^{2} ( x ) - 1 ) ^{2} cot ^{2} ( x )}{cot ^{2} ( x )} - \frac{1}{cot ^{2} ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{( sec ^{2} ( x ) - 1 ) ^{2} cot ^{2} ( x ) - 1}{cot ^{2} ( x )}

\frac{cot ^{2} ( x ) ( sec ^{2} ( x ) - 1 ) ^{2} - 1}{cot ^{2} ( x )} = 0

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

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

Rewrite using trig identities

- 1 + (- 1 + sec^{2} (x))^{2} cot^{2} (x)

sec^{2} (x) = tan^{2} (x) + 1

:فعّل نطريّة فيتاغوروس

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

= - 1 + (tan^{2} (x))^{2} cot^{2} (x)

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

(tan^{2} (x))^{2}

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

:فعّل قانون القوى

= tan^{2 \cdot 2} (x)

2 \cdot 2 = 4 :

اضرب الأعداد

= tan^{4} (x)

= - 1 + tan^{4} (x) cot^{2} (x)

- 1 + cot^{2} (x) tan^{4} (x) = 0

- 1 + cot^{2} (x) tan^{4} (x)

حلل إلى عوامل:

(tan^{2} (x) cot (x) + 1) (tan^{2} (x) cot (x) - 1)

- 1 + cot^{2} (x) tan^{4} (x)

- 1 + (cot (x) tan^{2} (x))^{2}

كـ

- 1 + cot^{2} (x) tan^{4} (x)

اكتب مجددًا

- 1 + cot^{2} (x) tan^{4} (x)

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

:فعّل قانون القوى

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

= - 1 + cot^{2} (x) (tan^{2} (x))^{2}

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

:فعّل قانون القوى

cot^{2} (x) (tan^{2} (x))^{2} = (cot (x) tan^{2} (x))^{2}

= - 1 + (cot (x) tan^{2} (x))^{2}

= - 1 + (cot (x) tan^{2} (x))^{2}

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

فعّل قانون فرق المربّعات

- 1 + (cot (x) tan^{2} (x))^{2} = (cot (x) tan^{2} (x) + 1) (cot (x) tan^{2} (x) - 1)

= (cot (x) tan^{2} (x) + 1) (cot (x) tan^{2} (x) - 1)

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

حلّ كل جزء على حدة

tan^{2} (x) cot (x) + 1 = 0 or tan^{2} (x) cot (x) - 1 = 0

tan^{2} (x) cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

tan^{2} (x) cot (x) + 1 = 0

Rewrite using trig identities

1 + cot (x) tan^{2} (x)

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

:Use the basic trigonometric identity

= 1 + cot (x) (\frac{1}{cot ( x )})^{2}

cot (x) (\frac{1}{cot ( x )})^{2} = \frac{1}{cot ( x )}

cot (x) (\frac{1}{cot ( x )})^{2}

(\frac{1}{cot ( x )})^{2} = \frac{1}{cot ^{2} ( x )}

(\frac{1}{cot ( x )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{cot ^{2} ( x )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{cot ^{2} ( x )}

= \frac{1}{cot ^{2} ( x )} cot (x)

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

:اضرب كسور

= \frac{1 \cdot cot ( x )}{cot ^{2} ( x )}

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

اضرب

= \frac{cot ( x )}{cot ^{2} ( x )}

cot (x) :

إلغ العوامل المشتركة

= \frac{1}{cot ( x )}

= 1 + \frac{1}{cot ( x )}

1 + \frac{1}{cot ( x )} = 0

cot (x)

اضرب الطرفين بـ

1 + \frac{1}{cot ( x )} = 0

cot (x)

اضرب الطرفين بـ

1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

بسّط

1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

1 \cdot cot (x)

بسّط:

cot (x)

1 \cdot cot (x)

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

اضرب

= cot (x)

\frac{1}{cot ( x )} cot (x)

بسّط:

1

\frac{1}{cot ( x )} cot (x)

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

:اضرب كسور

= \frac{1 \cdot cot ( x )}{cot ( x )}

cot (x) :

إلغ العوامل المشتركة

= 1

0 \cdot cot (x)

بسّط:

0

0 \cdot cot (x)

0 \cdot a = 0

فعّل القانون

= 0

cot (x) + 1 = 0

cot (x) + 1 = 0

cot (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

cot (x) + 1 = 0

من الطرفين

1

اطرح

cot (x) + 1 - 1 = 0 - 1

بسّط

cot (x) = - 1

cot (x) = - 1

افحص الإجبات

جد نقاط غير معرّفة:

cot (x) = 0

وقم بمساواتها لصفر

1 + \frac{1}{cot ( x )}

خذ المقامات في

cot (x) = 0

النقاط التالية غير معرّفة

cot (x) = 0

ضمّ النقاط غير المعرّفة مع الحلول

cot (x) = - 1

cot (x) = - 1 :

حلول عامّة لـ

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

tan^{2} (x) cot (x) - 1 = 0 : x = \frac{π}{4} + πn

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

Rewrite using trig identities

- 1 + cot (x) tan^{2} (x)

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

:Use the basic trigonometric identity

= - 1 + cot (x) (\frac{1}{cot ( x )})^{2}

cot (x) (\frac{1}{cot ( x )})^{2} = \frac{1}{cot ( x )}

cot (x) (\frac{1}{cot ( x )})^{2}

(\frac{1}{cot ( x )})^{2} = \frac{1}{cot ^{2} ( x )}

(\frac{1}{cot ( x )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{cot ^{2} ( x )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{cot ^{2} ( x )}

= \frac{1}{cot ^{2} ( x )} cot (x)

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

:اضرب كسور

= \frac{1 \cdot cot ( x )}{cot ^{2} ( x )}

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

اضرب

= \frac{cot ( x )}{cot ^{2} ( x )}

cot (x) :

إلغ العوامل المشتركة

= \frac{1}{cot ( x )}

= - 1 + \frac{1}{cot ( x )}

- 1 + \frac{1}{cot ( x )} = 0

cot (x)

اضرب الطرفين بـ

- 1 + \frac{1}{cot ( x )} = 0

cot (x)

اضرب الطرفين بـ

- 1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

بسّط

- 1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

- 1 \cdot cot (x)

بسّط:

- cot (x)

- 1 \cdot cot (x)

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

اضرب

= - cot (x)

\frac{1}{cot ( x )} cot (x)

بسّط:

1

\frac{1}{cot ( x )} cot (x)

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

:اضرب كسور

= \frac{1 \cdot cot ( x )}{cot ( x )}

cot (x) :

إلغ العوامل المشتركة

= 1

0 \cdot cot (x)

بسّط:

0

0 \cdot cot (x)

0 \cdot a = 0

فعّل القانون

= 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

- cot (x) + 1 = 0

من الطرفين

1

اطرح

- cot (x) + 1 - 1 = 0 - 1

بسّط

- cot (x) = - 1

- cot (x) = - 1

- 1

اقسم الطرفين على

- cot (x) = - 1

- 1

اقسم الطرفين على

\frac{- cot ( x )}{- 1} = \frac{- 1}{- 1}

بسّط

cot (x) = 1

cot (x) = 1

افحص الإجبات

جد نقاط غير معرّفة:

cot (x) = 0

وقم بمساواتها لصفر

- 1 + \frac{1}{cot ( x )}

خذ المقامات في

cot (x) = 0

النقاط التالية غير معرّفة

cot (x) = 0

ضمّ النقاط غير المعرّفة مع الحلول

cot (x) = 1

cot (x) = 1 :

حلول عامّة لـ

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

وحّد الحلول

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

تأكّد من صحّة الحلول عن طريق تعويضها في المعادلة الأصليّة

للتحقّق من دقّة الحلول

sec^{2} (x) - 1 = \frac{1}{cot ( x )}

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

\frac{3 π}{4} + πn

افحص الحل:

خطأ

\frac{3 π}{4} + πn

n = 1

استبدل

\frac{3 π}{4} + π 1

x = \frac{3 π}{4} + π 1

عوّض

, sec^{2} (x) - 1 = \frac{1}{cot ( x )}

في

sec^{2} (\frac{3 π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{3 π}{4} + π 1 )}

بسّط

1 = - 1

\Rightarrow خطأ

\frac{π}{4} + πn

افحص الحل:

صحيح

\frac{π}{4} + πn

n = 1

استبدل

\frac{π}{4} + π 1

x = \frac{π}{4} + π 1

عوّض

, sec^{2} (x) - 1 = \frac{1}{cot ( x )}

في

sec^{2} (\frac{π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{π}{4} + π 1 )}

بسّط

1 = 1

\Rightarrow صحيح

x = \frac{π}{4} + πn

رسم

أمثلة شائعة

4cos(2x)-3cos(x)+1=0

4 cos (2 x) - 3 cos (x) + 1 = 0

2tan^4(x)-tan^2(x)-15=0

2 tan^{4} (x) - tan^{2} (x) - 15 = 0

cos(x)=-sqrt(1/2)

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

6cos(θ)=-6(1+cos(θ))

6 cos (θ) = - 6 (1 + cos (θ))

2sin^2(x)+5sin(x)+3=0

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