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شائع علم المثلثات >

2sec(x)=tan(x)+cot(x)

الحلّ

2 sec (x) = tan (x) + cot (x)

الحلّ

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

درجات

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

2 sec (x) = tan (x) + cot (x)

من الطرفين

tan (x) + cot (x)

اطرح

2 sec (x) - tan (x) - cot (x) = 0

sin, cos :

عبّر بواسطة

- cot (x) - tan (x) + 2 sec (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

2 \cdot \frac{1}{cos ( x )} = \frac{2}{cos ( x )}

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

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

:اضرب كسور

= \frac{1 \cdot 2}{cos ( x )}

1 \cdot 2 = 2 :

اضرب الأعداد

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

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

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

وحّد الكسور:

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

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

فعّل القانون

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

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

sin (x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= 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 ) + 2}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

2 sin (x) - 1

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

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

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

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

افتح أقواس

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

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

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

sin (x) (2 - sin (x))

وسٌع:

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

sin (x) (2 - sin (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = sin (x), b = 2, c = sin (x)

= sin (x) \cdot 2 - sin (x) sin (x)

= 2 sin (x) - sin (x) sin (x)

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

sin (x) sin (x)

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

:فعّل قانون القوى

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

= sin^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (x)

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

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

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

بسّط:

2 sin (x) - 1

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

جمّع التعابير المتشابهة

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

sin^{2} (x) - sin^{2} (x) = 0 :

اجمع العناصر المتشابهة

= 2 sin (x) - 1

= 2 sin (x) - 1

= 2 sin (x) - 1

- 1 + 2 sin (x) = 0

انقل

1

إلى الجانب الأيمن

- 1 + 2 sin (x) = 0

للطرفين

1

أضف

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

بسّط

2 sin (x) = 1

2 sin (x) = 1

2

اقسم الطرفين على

2 sin (x) = 1

2

اقسم الطرفين على

\frac{2 sin ( x )}{2} = \frac{1}{2}

بسّط

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2} :

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

رسم

أمثلة شائعة

sqrt(3)sin(2x)=cos(2x)

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

2sin(6x)+sqrt(3)=0

2 sin (6 x) + 3 = 0

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

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

cos(x)= 6/20

cos (x) = \frac{6}{20}

2sin(x-60)=cos(x-30)

2 sin (x - 6 0^{\circ}) = cos (x - 3 0^{\circ})