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شائع علم المثلثات >

(sec(x)+csc(x))/(1+tan(x))=sin(x)

الحلّ

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} = sin (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} = sin (x)

من الطرفين

sin (x)

اطرح

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x) = 0

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x)

بسّط:

\frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x)

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

:حوّل الأعداد لكسور

= \frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - \frac{sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

\frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )} = 0

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

sec (x) + csc (x) - sin (x) (1 + tan (x)) = 0

sin, cos :

عبّر بواسطة

csc (x) + sec (x) - (1 + tan (x)) sin (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

اضرب

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

sin (x), cos (x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x), cos (x)

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{1}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

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

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

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

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

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

حلل إلى عوامل:

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

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

أعد الكتابة كـ

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

(cos (x) + sin (x))

قم باخراج العامل المشترك

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

- sin^{2} (x) + 1

حلل إلى عوامل:

- (sin (x) + 1) (sin (x) - 1)

- sin^{2} (x) + 1

- 1

قم باخراج العامل المشترك

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

sin^{2} (x) - 1

حلل إلى عوامل:

(sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

1^{2}

كـ

1

اكتب مجددًا

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

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

فعّل قانون فرق المربّعات

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

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

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

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

بسّط

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

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

حلّ كل جزء على حدة

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

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) + sin (x) = 0

Rewrite using trig identities

cos (x) + sin (x) = 0

cos (x) \neq = 0, cos (x)

اقسم الطرفين على

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

بسّط

1 + \frac{sin ( x )}{cos ( x )} = 0

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

:Use the basic trigonometric identity

1 + tan (x) = 0

1 + tan (x) = 0

انقل

1

إلى الجانب الأيمن

1 + tan (x) = 0

من الطرفين

1

اطرح

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

بسّط

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1 :

حلول عامّة لـ

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

sin (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

sin (x) + 1 = 0

من الطرفين

1

اطرح

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

بسّط

sin (x) = - 1

sin (x) = - 1

sin (x) = - 1 :

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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

sin (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

sin (x) - 1 = 0

للطرفين

1

أضف

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

بسّط

sin (x) = 1

sin (x) = 1

sin (x) = 1 :

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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

وحّد الحلول

x = \frac{3 π}{4} + πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

\frac{3 π}{4} + πn, \frac{3 π}{2} + 2 πn, \frac{π}{2} + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

sec(x)-(sin(x))/(cos(x))=cos(x)

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

cos(θ)=0.4,sec(θ)

cos (θ) = 0.4, sec (θ)

cot(θ)=(1+cos^2(θ))/(2sin(θ)cos(θ))

cot (θ) = \frac{1 + cos ^{2} ( θ )}{2 sin ( θ ) cos ( θ )}

sec(3x)-csc(30)=0,(x+35)/5

sec (3 x) - csc (3 0^{\circ}) = 0, \frac{x + 35}{5}

4cos(2θ)-10cos(θ)+14=7,0<= θ<360

4 cos (2 θ) - 10 cos (θ) + 14 = 7, 0 \leq θ < 36 0^{\circ}