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شائع علم المثلثات >

prove (cot(3θ)-tan(3θ))/(sin(3θ)+cos(3θ))=csc(3θ)-sec(3θ)

الحلّ

prove

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )} = csc (3 θ) - sec (3 θ)

الحلّ

صحيح

خطوات الحلّ

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )} = csc (3 θ) - sec (3 θ)

قم بالعمل على الطرف الأيسر

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )}

sin, cos :

عبّر بواسطة

\frac{cot ( 3 θ ) - tan ( 3 θ )}{cos ( 3 θ ) + sin ( 3 θ )}

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

:Use the basic trigonometric identity

= \frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - tan ( 3 θ )}{cos ( 3 θ ) + sin ( 3 θ )}

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

:Use the basic trigonometric identity

= \frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - \frac{s i n ( 3 θ )}{c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

\frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - \frac{s i n ( 3 θ )}{c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

بسّط:

\frac{cos ( 3 θ ) - sin ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

\frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - \frac{s i n ( 3 θ )}{c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

\frac{cos ( 3 θ )}{sin ( 3 θ )} - \frac{sin ( 3 θ )}{cos ( 3 θ )}

وحّد:

\frac{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

\frac{cos ( 3 θ )}{sin ( 3 θ )} - \frac{sin ( 3 θ )}{cos ( 3 θ )}

sin (3 θ), cos (3 θ)

المضاعف المشترك الأصغر لـ:

sin (3 θ) cos (3 θ)

sin (3 θ), cos (3 θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (3 θ)

cos (3 θ)

= sin (3 θ) cos (3 θ)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (3 θ) cos (3 θ)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( 3 θ )}{sin ( 3 θ )} :

multiply the denominator and numerator by

cos (3 θ)

\frac{cos ( 3 θ )}{sin ( 3 θ )} = \frac{cos ( 3 θ ) cos ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )} = \frac{cos ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

For

\frac{sin ( 3 θ )}{cos ( 3 θ )} :

multiply the denominator and numerator by

sin (3 θ)

\frac{sin ( 3 θ )}{cos ( 3 θ )} = \frac{sin ( 3 θ ) sin ( 3 θ )}{cos ( 3 θ ) sin ( 3 θ )} = \frac{sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{cos ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )} - \frac{sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{\frac{c o s ^{2} ( 3 θ ) - s i n ^{2} ( 3 θ )}{s i n ( 3 θ ) c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

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

: استخدم ميزات الكسور التالية

= \frac{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ ) ( cos ( 3 θ ) + sin ( 3 θ ) )}

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

cos^{2} (3 θ) - sin^{2} (3 θ) = (cos (3 θ) + sin (3 θ)) (cos (3 θ) - sin (3 θ))

= \frac{( cos ( 3 θ ) + sin ( 3 θ ) ) ( cos ( 3 θ ) - sin ( 3 θ ) )}{sin ( 3 θ ) cos ( 3 θ ) ( cos ( 3 θ ) + sin ( 3 θ ) )}

cos (3 θ) + sin (3 θ) :

إلغ العوامل المشتركة

= \frac{cos ( 3 θ ) - sin ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{cos ( 3 θ ) - sin ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{cos ( 3 θ ) - sin ( 3 θ )}{cos ( 3 θ ) sin ( 3 θ )}

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{cos ( 3 θ ) - \frac{1}{c s c ( 3 θ )}}{cos ( 3 θ ) \frac{1}{c s c ( 3 θ )}}

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

:Use the basic trigonometric identity

\frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ )} \cdot \frac{1}{c s c ( 3 θ )}}

بسّط

\frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ )} \cdot \frac{1}{c s c ( 3 θ )}}

\frac{1}{sec ( 3 θ )} \cdot \frac{1}{csc ( 3 θ )}

اضرب بـ:

\frac{1}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{sec ( 3 θ )} \cdot \frac{1}{csc ( 3 θ )}

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

:اضرب كسور

= \frac{1 \cdot 1}{sec ( 3 θ ) csc ( 3 θ )}

1 \cdot 1 = 1 :

اضرب الأعداد

= \frac{1}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ ) c s c ( 3 θ )}}

\frac{1}{sec ( 3 θ )} - \frac{1}{csc ( 3 θ )}

وحّد:

\frac{csc ( 3 θ ) - sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{sec ( 3 θ )} - \frac{1}{csc ( 3 θ )}

sec (3 θ), csc (3 θ)

المضاعف المشترك الأصغر لـ:

sec (3 θ) csc (3 θ)

sec (3 θ), csc (3 θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sec (3 θ)

csc (3 θ)

= sec (3 θ) csc (3 θ)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sec (3 θ) csc (3 θ)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sec ( 3 θ )} :

multiply the denominator and numerator by

csc (3 θ)

\frac{1}{sec ( 3 θ )} = \frac{1 \cdot csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )} = \frac{csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

For

\frac{1}{csc ( 3 θ )} :

multiply the denominator and numerator by

sec (3 θ)

\frac{1}{csc ( 3 θ )} = \frac{1 \cdot sec ( 3 θ )}{csc ( 3 θ ) sec ( 3 θ )} = \frac{sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )} - \frac{sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{csc ( 3 θ ) - sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{\frac{c s c ( 3 θ ) - s e c ( 3 θ )}{s e c ( 3 θ ) c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ ) c s c ( 3 θ )}}

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

:اقسم الكسور

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) sec ( 3 θ ) csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ ) \cdot 1}

بسّط

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) sec ( 3 θ ) csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

sec (3 θ) :

إلغ العوامل المشتركة

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) csc ( 3 θ )}{csc ( 3 θ )}

csc (3 θ) :

إلغ العوامل المشتركة

= csc (3 θ) - sec (3 θ)

csc (3 θ) - sec (3 θ)

csc (3 θ) - sec (3 θ)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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