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شائع علم المثلثات >

(tan(x)+cos(x))/((1+sin(x)))= 1/(cos(x))

الحلّ

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

الحلّ

x = 2 πn, x = π + 2 πn

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

\frac{1}{cos ( x )}

اطرح

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

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

بسّط:

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

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

1 + sin (x), cos (x)

المضاعف المشترك الأصغر لـ:

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

1 + sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

1 + sin (x)

cos (x)

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

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

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x) + 1

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

وسٌع:

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

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

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

cos (x) (tan (x) + cos (x))

وسٌع:

cos (x) tan (x) + cos^{2} (x)

cos (x) (tan (x) + cos (x))

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = cos (x), b = tan (x), c = cos (x)

= cos (x) tan (x) + cos (x) cos (x)

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

cos (x) cos (x)

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

:فعّل قانون القوى

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

= cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (x)

= cos (x) tan (x) + cos^{2} (x)

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

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

- (sin (x) + 1)

افتح أقواس

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

فعّل قوانين سالب-موجب

+ (- a) = - a

= - sin (x) - 1

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

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

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

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

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

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

بسّط:

- 1 + cos^{2} (x)

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

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

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

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

= sin (x)

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

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

اجمع العناصر المتشابهة

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

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

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

بالاستعانة بطريقة التعويض

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

cos (x) = u :

على افتراض أنّ

- 1 + u^{2} = 0

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

انقل

1

إلى الجانب الأيمن

- 1 + u^{2} = 0

للطرفين

1

أضف

- 1 + u^{2} + 1 = 0 + 1

بسّط

u^{2} = 1

u^{2} = 1

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 1, u = - 1

1 = 1

1

1 = 1

فعّل القانون

= 1

- 1 = - 1

- 1

1 = 1

فعّل القانون

= - 1

u = 1, u = - 1

u = cos (x)

استبدل مجددًا

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (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}

x = 0 + 2 πn

x = 0 + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (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}

x = π + 2 πn

x = π + 2 πn

وحّد الحلول

x = 2 πn, x = π + 2 πn

رسم

أمثلة شائعة

sin(θ)= 4/5 ,0<= θ<= pi/2

sin (θ) = \frac{4}{5}, 0 \leq θ \leq \frac{π}{2}

((tan(x)+2sin(x)))/((tan(x)-2sin(x)))=3

\frac{( tan ( x ) + 2 sin ( x ) )}{( tan ( x ) - 2 sin ( x ) )} = 3

1-sin(2x)=0

1 - sin (2 x) = 0

2cos(3x)+cos(2x)+1=0

2 cos (3 x) + cos (2 x) + 1 = 0

sin(x)=sin(pi/5)

sin (x) = sin (\frac{π}{5})