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شائع علم المثلثات >

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

الحلّ

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

الحلّ

x = \frac{π}{4} + πn

درجات

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

خطوات الحلّ

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

من الطرفين

1

اطرح

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

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

بسّط:

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

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

1 = \frac{1}{1}

:حوّل الأعداد لكسور

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

csc (x), sec (x), 1

المضاعف المشترك الأصغر لـ:

csc (x) sec (x)

csc (x), sec (x), 1

Lowest Common Multiplier (LCM)

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

= csc (x) sec (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

csc (x) sec (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sec (x)

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

For

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

multiply the denominator and numerator by

csc (x)

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

For

\frac{1}{1} :

multiply the denominator and numerator by

csc (x) sec (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

sin (x) :

إلغ العوامل المشتركة

= 1

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

جمّع التعابير المتشابهة

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

1 + 1 = 2 :

اجمع الأعداد

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

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

:حوّل الأعداد لكسور

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

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

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

= - 1 + sin (2 x)

- 1 + sin (2 x) = 0

انقل

1

إلى الجانب الأيمن

- 1 + sin (2 x) = 0

للطرفين

1

أضف

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

بسّط

sin (2 x) = 1

sin (2 x) = 1

sin (2 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}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

حلّ:

x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

2

اقسم الطرفين على

2 x = \frac{π}{2} + 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

بسّط

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

بسّط:

\frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{π}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

رسم

أمثلة شائعة

cot(θ)=-8

cot (θ) = - 8

solvefor t,0=(1.8)cos(2pift)

so l v e f or t, 0 = (1.8) cos (2 π f t)

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

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

tan(θ)-7=-12cot(θ)

tan (θ) - 7 = - 12 cot (θ)

4cos^2(3x)-1=0

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