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شائع علم المثلثات >

3/(cot(x))=8-5tan(x)

الحلّ

\frac{3}{cot ( x )} = 8 - 5 tan (x)

الحلّ

x = \frac{π}{4} + πn

+1

درجات

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

خطوات الحلّ

\frac{3}{cot ( x )} = 8 - 5 tan (x)

من الطرفين

8 - 5 tan (x)

اطرح

\frac{3}{cot ( x )} - 8 + 5 tan (x) = 0

\frac{3}{cot ( x )} - 8 + 5 tan (x)

بسّط:

\frac{3 - 8 cot ( x ) + 5 tan ( x ) cot ( x )}{cot ( x )}

\frac{3}{cot ( x )} - 8 + 5 tan (x)

8 = \frac{8 cot ( x )}{cot ( x )}, 5 tan (x) = \frac{5 tan ( x ) cot ( x )}{cot ( x )}

:حوّل الأعداد لكسور

= \frac{3}{cot ( x )} - \frac{8 cot ( x )}{cot ( x )} + \frac{5 tan ( x ) cot ( x )}{cot ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{3 - 8 cot ( x ) + 5 tan ( x ) cot ( x )}{cot ( x )}

\frac{3 - 8 cot ( x ) + 5 tan ( x ) cot ( x )}{cot ( x )} = 0

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

3 - 8 cot (x) + 5 tan (x) cot (x) = 0

Rewrite using trig identities

3 - 8 cot (x) + 5 cot (x) tan (x)

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

:Use the basic trigonometric identity

= 3 - 8 cot (x) + 5 cot (x) \frac{1}{cot ( x )}

3 - 8 cot (x) + 5 cot (x) \frac{1}{cot ( x )}

بسّط:

- 8 cot (x) + 8

3 - 8 cot (x) + 5 cot (x) \frac{1}{cot ( x )}

5 cot (x) \frac{1}{cot ( x )} = 5

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

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

:اضرب كسور

= \frac{1 \cdot 5 cot ( x )}{cot ( x )}

cot (x) :

إلغ العوامل المشتركة

= 1 \cdot 5

1 \cdot 5 = 5 :

اضرب الأعداد

= 5

= 3 - 8 cot (x) + 5

جمّع التعابير المتشابهة

= - 8 cot (x) + 3 + 5

3 + 5 = 8 :

اجمع الأعداد

= - 8 cot (x) + 8

= - 8 cot (x) + 8

8 - 8 cot (x) = 0

انقل

8

إلى الجانب الأيمن

8 - 8 cot (x) = 0

من الطرفين

8

اطرح

8 - 8 cot (x) - 8 = 0 - 8

بسّط

- 8 cot (x) = - 8

- 8 cot (x) = - 8

- 8

اقسم الطرفين على

- 8 cot (x) = - 8

- 8

اقسم الطرفين على

\frac{- 8 cot ( x )}{- 8} = \frac{- 8}{- 8}

بسّط

cot (x) = 1

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

رسم

أمثلة شائعة

sin(θ)= 3/8 ,sin(2θ)

sin (θ) = \frac{3}{8}, sin (2 θ)

2(1-cos^2(x))= 3/2

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

cos(3θ)=-(sqrt(2))/2

cos (3 θ) = - \frac{2}{2}

25*sin(x)-1.5*cos(x)=20

25 \cdot sin (x) - 1.5 \cdot cos (x) = 20

3cos(3x)=2cos(x)

3 cos (3 x) = 2 cos (x)