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شائع علم المثلثات >

(cos(x))/(sec(x))-(sin(x))/(csc(x))=1

الحلّ

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

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

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

من الطرفين

1

اطرح

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

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

بسّط:

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

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

1 = \frac{1}{1}

:حوّل الأعداد لكسور

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

sec (x), csc (x), 1

المضاعف المشترك الأصغر لـ:

sec (x) csc (x)

sec (x), csc (x), 1

Lowest Common Multiplier (LCM)

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

= sec (x) csc (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sec (x) csc (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

csc (x)

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

For

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

multiply the denominator and numerator by

sec (x)

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

For

\frac{1}{1} :

multiply the denominator and numerator by

sec (x) csc (x)

\frac{1}{1} = \frac{1 \cdot sec ( x ) csc ( x )}{1 \cdot sec ( x ) csc ( x )} = \frac{sec ( x ) csc ( x )}{sec ( x ) csc ( x )}

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

\frac{cos ( x ) csc ( x ) - sin ( x ) sec ( x ) - sec ( x ) csc ( x )}{sec ( x ) csc ( x )} = 0

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

cos (x) csc (x) - sin (x) sec (x) - sec (x) csc (x) = 0

sin, cos :

عبّر بواسطة

cos (x) csc (x) - csc (x) sec (x) - sec (x) sin (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

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

اضرب

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

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

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

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

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

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

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

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

:اضرب كسور

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

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

اضرب

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

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

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

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (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{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 )}

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

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

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

= - 1 + cos (2 x)

- 1 + cos (2 x) = 0

انقل

1

إلى الجانب الأيمن

- 1 + cos (2 x) = 0

للطرفين

1

أضف

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

بسّط

cos (2 x) = 1

cos (2 x) = 1

cos (2 x) = 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}

2 x = 0 + 2 πn

2 x = 0 + 2 πn

2 x = 0 + 2 πn

حلّ:

x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

2

اقسم الطرفين على

2 x = 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{2 πn}{2}

بسّط

x = πn

x = πn

x = πn

πn :

بما أنّ المعادلة غير معرّفة لـ

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

tan(x)= 12/35

tan (x) = \frac{12}{35}

4=sec(θ)

4 = sec (θ)

1+sqrt(2)sin(x)=cos(2x)

1 + 2 sin (x) = cos (2 x)

12-sin(θ)=cos(2θ)

12 - sin (θ) = cos (2 θ)

2cos(θ)=3

2 cos (θ) = 3