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شائع علم المثلثات >

tan(x)+2cos(x)csc(x)=sec(x)cos(x)+cot(x)

الحلّ

tan (x) + 2 cos (x) csc (x) = sec (x) cos (x) + cot (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

tan (x) + 2 cos (x) csc (x) = sec (x) cos (x) + cot (x)

من الطرفين

sec (x) cos (x) + cot (x)

اطرح

tan (x) + 2 cos (x) csc (x) - sec (x) cos (x) - cot (x) = 0

sin, cos :

عبّر بواسطة

- cot (x) + tan (x) - cos (x) sec (x) + 2 cos (x) csc (x)

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

:Use the basic trigonometric identity

= - \frac{cos ( x )}{sin ( x )} + tan (x) - cos (x) sec (x) + 2 cos (x) csc (x)

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

:Use the basic trigonometric identity

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

sec (x) = \frac{1}{cos ( x )}

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

= 1

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

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

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

:اضرب كسور

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

1 \cdot 2 = 2 :

اضرب الأعداد

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

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

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

وحّد الكسور:

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

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

فعّل القانون

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

- cos (x) + 2 cos (x) = cos (x) :

اجمع العناصر المتشابهة

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

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

1 = \frac{1}{1}

:حوّل الأعداد لكسور

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

sin (x), cos (x), 1

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x), 1

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= sin (x) cos (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) cos (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( x )}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

For

\frac{sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

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

For

\frac{1}{1} :

multiply the denominator and numerator by

sin (x) cos (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

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

Rewrite using trig identities

- cos (x) sin (x) + 1

2 sin (x) cos (x) = sin (2 x)

:فعّل متطابقة الزاوية المضاعفة

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

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

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

انقل

1

إلى الجانب الأيمن

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

من الطرفين

1

اطرح

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

بسّط

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

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

2

اضرب الطرفين بـ

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

2

اضرب الطرفين بـ

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

بسّط

- sin (2 x) = - 2

- sin (2 x) = - 2

- 1

اقسم الطرفين على

- sin (2 x) = - 2

- 1

اقسم الطرفين على

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

بسّط

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

cos(x)=1.22719

cos (x) = 1.22719

cos(a)=(sqrt(5))/5

cos (a) = \frac{5}{5}

5sec^2(x)-5=0,0<= x<2pi

5 sec^{2} (x) - 5 = 0, 0 \leq x < 2 π

2sin(5x)+3=2

2 sin (5 x) + 3 = 2

5sin(y)-5=3sin(y)-4

5 sin (y) - 5 = 3 sin (y) - 4