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شائع علم المثلثات >

1/(tan(α))+tan(α)= 1/(sin(α))

الحلّ

\frac{1}{tan ( α )} + tan (α) = \frac{1}{sin ( α )}

الحلّ

α \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{1}{tan ( α )} + tan (α) = \frac{1}{sin ( α )}

من الطرفين

\frac{1}{sin ( α )}

اطرح

\frac{1}{tan ( α )} + tan (α) - \frac{1}{sin ( α )} = 0

\frac{1}{tan ( α )} + tan (α) - \frac{1}{sin ( α )}

بسّط:

\frac{sin ( α ) + tan ^{2} ( α ) sin ( α ) - tan ( α )}{tan ( α ) sin ( α )}

\frac{1}{tan ( α )} + tan (α) - \frac{1}{sin ( α )}

tan (α) = \frac{tan ( α )}{1}

:حوّل الأعداد لكسور

= \frac{1}{tan ( α )} + \frac{tan ( α )}{1} - \frac{1}{sin ( α )}

tan (α), 1, sin (α)

المضاعف المشترك الأصغر لـ:

tan (α) sin (α)

tan (α), 1, sin (α)

Lowest Common Multiplier (LCM)

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

= tan (α) sin (α)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

tan (α) sin (α)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{tan ( α )} :

multiply the denominator and numerator by

sin (α)

\frac{1}{tan ( α )} = \frac{1 \cdot sin ( α )}{tan ( α ) sin ( α )} = \frac{sin ( α )}{tan ( α ) sin ( α )}

For

\frac{tan ( α )}{1} :

multiply the denominator and numerator by

tan (α) sin (α)

\frac{tan ( α )}{1} = \frac{tan ( α ) tan ( α ) sin ( α )}{1 \cdot tan ( α ) sin ( α )} = \frac{tan ^{2} ( α ) sin ( α )}{tan ( α ) sin ( α )}

For

\frac{1}{sin ( α )} :

multiply the denominator and numerator by

tan (α)

\frac{1}{sin ( α )} = \frac{1 \cdot tan ( α )}{sin ( α ) tan ( α )} = \frac{tan ( α )}{tan ( α ) sin ( α )}

= \frac{sin ( α )}{tan ( α ) sin ( α )} + \frac{tan ^{2} ( α ) sin ( α )}{tan ( α ) sin ( α )} - \frac{tan ( α )}{tan ( α ) sin ( α )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ( α ) + tan ^{2} ( α ) sin ( α ) - tan ( α )}{tan ( α ) sin ( α )}

\frac{sin ( α ) + tan ^{2} ( α ) sin ( α ) - tan ( α )}{tan ( α ) sin ( α )} = 0

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

sin (α) + tan^{2} (α) sin (α) - tan (α) = 0

sin, cos :

عبّر بواسطة

sin (α) - tan (α) + sin (α) tan^{2} (α)

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

:Use the basic trigonometric identity

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

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

بسّط:

\frac{cos ^{2} ( α ) sin ( α ) - sin ( α ) cos ( α ) + sin ^{3} ( α )}{cos ^{2} ( α )}

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

sin (α) (\frac{sin ( α )}{cos ( α )})^{2} = \frac{sin ^{3} ( α )}{cos ^{2} ( α )}

sin (α) (\frac{sin ( α )}{cos ( α )})^{2}

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

(\frac{sin ( α )}{cos ( α )})^{2}

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

:فعّل قانون القوى

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

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

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

:اضرب كسور

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

sin^{2} (α) sin (α) = sin^{3} (α)

sin^{2} (α) sin (α)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

sin^{2} (α) sin (α) = sin^{2 + 1} (α)

= sin^{2 + 1} (α)

2 + 1 = 3 :

اجمع الأعداد

= sin^{3} (α)

= \frac{sin ^{3} ( α )}{cos ^{2} ( α )}

= sin (α) - \frac{sin ( α )}{cos ( α )} + \frac{sin ^{3} ( α )}{cos ^{2} ( α )}

sin (α) = \frac{sin ( α )}{1}

:حوّل الأعداد لكسور

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

1, cos (α), cos^{2} (α)

المضاعف المشترك الأصغر لـ:

cos^{2} (α)

1, cos (α), cos^{2} (α)

Lowest Common Multiplier (LCM)

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

= cos^{2} (α)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos^{2} (α)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{sin ( α )}{1} :

multiply the denominator and numerator by

cos^{2} (α)

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

For

\frac{sin ( α )}{cos ( α )} :

multiply the denominator and numerator by

cos (α)

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

= \frac{sin ( α ) cos ^{2} ( α )}{cos ^{2} ( α )} - \frac{sin ( α ) cos ( α )}{cos ^{2} ( α )} + \frac{sin ^{3} ( α )}{cos ^{2} ( α )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ( α ) cos ^{2} ( α ) - sin ( α ) cos ( α ) + sin ^{3} ( α )}{cos ^{2} ( α )}

= \frac{cos ^{2} ( α ) sin ( α ) - sin ( α ) cos ( α ) + sin ^{3} ( α )}{cos ^{2} ( α )}

\frac{sin ^{3} ( α ) - cos ( α ) sin ( α ) + cos ^{2} ( α ) sin ( α )}{cos ^{2} ( α )} = 0

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

sin^{3} (α) - cos (α) sin (α) + cos^{2} (α) sin (α) = 0

sin^{3} (α) - cos (α) sin (α) + cos^{2} (α) sin (α)

حلل إلى عوامل:

sin (α) (sin^{2} (α) - cos (α) + cos^{2} (α))

sin^{3} (α) - cos (α) sin (α) + cos^{2} (α) sin (α)

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

sin^{3} (α) = sin (α) sin^{2} (α)

= sin (α) sin^{2} (α) - sin (α) cos (α) + sin (α) cos^{2} (α)

sin (α)

قم باخراج العامل المشترك

= sin (α) (sin^{2} (α) - cos (α) + cos^{2} (α))

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

Rewrite using trig identities

sin (α) (sin^{2} (α) - cos (α) + cos^{2} (α))

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

= sin (α) (- cos (α) + 1)

sin (α) (- cos (α) + 1) = 0

حلّ كل جزء على حدة

sin (α) = 0 or - cos (α) + 1 = 0

sin (α) = 0 : α = 2 πn, α = π + 2 πn

sin (α) = 0

sin (α) = 0 :

حلول عامّة لـ

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

α = 0 + 2 πn, α = π + 2 πn

α = 0 + 2 πn, α = π + 2 πn

α = 0 + 2 πn

حلّ:

α = 2 πn

α = 0 + 2 πn

0 + 2 πn = 2 πn

α = 2 πn

α = 2 πn, α = π + 2 πn

- cos (α) + 1 = 0 : α = 2 πn

- cos (α) + 1 = 0

انقل

1

إلى الجانب الأيمن

- cos (α) + 1 = 0

من الطرفين

1

اطرح

- cos (α) + 1 - 1 = 0 - 1

بسّط

- cos (α) = - 1

- cos (α) = - 1

- 1

اقسم الطرفين على

- cos (α) = - 1

- 1

اقسم الطرفين على

\frac{- cos ( α )}{- 1} = \frac{- 1}{- 1}

بسّط

cos (α) = 1

cos (α) = 1

cos (α) = 1 :

حلول عامّة لـ

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

α = 0 + 2 πn

α = 0 + 2 πn

α = 0 + 2 πn

حلّ:

α = 2 πn

α = 0 + 2 πn

0 + 2 πn = 2 πn

α = 2 πn

α = 2 πn

وحّد الحلول

α = 2 πn, α = π + 2 πn

2 πn, π + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

α \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

8sin^2(x)-1=5

8 sin^{2} (x) - 1 = 5

csc(θ)= 17/8

csc (θ) = \frac{17}{8}

cos(x)= 60/61 ,cos(2x)

cos (x) = \frac{60}{61}, cos (2 x)

cos(b)=0.47

cos (b) = 0.47

tan^2(x)-4sec(x)=4

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