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شائع علم المثلثات >

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

الحلّ

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

الحلّ

x = \frac{π}{4} + πn

درجات

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

خطوات الحلّ

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

من الطرفين

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

اطرح

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

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

بسّط:

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

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

sin (x), tan (x)

المضاعف المشترك الأصغر لـ:

sin (x) tan (x)

sin (x), tan (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

tan (x)

= sin (x) tan (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) tan (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

tan (x)

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

For

\frac{2}{tan ( x )} :

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

وسٌع:

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

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

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

وسٌع:

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

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

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

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

= sin (x) cos (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)

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

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

sin (x) cos (x) - 2 sin (x) cos (x) = - sin (x) cos (x) :

اجمع العناصر المتشابهة

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

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

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

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

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

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

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

حلل إلى عوامل:

sin (x) (sin (x) - cos (x))

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

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

:فعّل قانون القوى

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

= sin (x) sin (x) - sin (x) cos (x)

sin (x)

قم باخراج العامل المشترك

= sin (x) (sin (x) - cos (x))

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

حلّ كل جزء على حدة

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0 :

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

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

sin (x) - cos (x) = 0

Rewrite using trig identities

sin (x) - cos (x) = 0

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

tan (x) - 1 = 0

tan (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

tan (x) - 1 = 0

للطرفين

1

أضف

tan (x) - 1 + 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{π}{4} + πn

x = \frac{π}{4} + πn

وحّد الحلول

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

2 πn, π + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

x = \frac{π}{4} + πn

رسم

أمثلة شائعة

3tan(x+45)=sqrt(3)

3 tan (x + 4 5^{\circ}) = 3

0=sin(pix)

0 = sin (π x)

sec(θ)csc(θ)=1

sec (θ) csc (θ) = 1

prove cos(a+270)=sin(a)

p ro v e cos (a + 27 0^{\circ}) = sin (a)

1+sin(θ)=2-sin(θ)

1 + sin (θ) = 2 - sin (θ)