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شائع علم المثلثات >

tan(3x+26)=cot(5x)

الحلّ

tan (3 x + 2 6^{\circ}) = cot (5 x)

الحلّ

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}, x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

راديان

x = \frac{2 π}{45} + \frac{π}{4} n, x = \frac{61 π}{360} + \frac{π}{4} n

خطوات الحلّ

tan (3 x + 2 6^{\circ}) = cot (5 x)

من الطرفين

cot (5 x)

اطرح

tan (3 x + 2 6^{\circ}) - cot (5 x) = 0

sin, cos :

عبّر بواسطة

- cot (5 x) + tan (2 6^{\circ} + 3 x)

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

:Use the basic trigonometric identity

= - \frac{cos ( 5 x )}{sin ( 5 x )} + tan (2 6^{\circ} + 3 x)

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

:Use the basic trigonometric identity

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

بسّط:

\frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

\frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )} = \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

\frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( 2 6 ^{\circ} + 3 x )}

2 6^{\circ} + 3 x

وحّد:

\frac{234 0 ^{\circ} + 270 x}{90}

2 6^{\circ} + 3 x

3 x = \frac{3 x 90}{90}

:حوّل الأعداد لكسور

= 2 6^{\circ} + \frac{3 x \cdot 90}{90}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{234 0 ^{\circ} + 3 x \cdot 90}{90}

3 \cdot 90 = 270 :

اضرب الأعداد

= \frac{234 0 ^{\circ} + 270 x}{90}

= \frac{sin ( 2 6 ^{\circ} + 3 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

2 6^{\circ} + 3 x

وحّد:

\frac{234 0 ^{\circ} + 270 x}{90}

2 6^{\circ} + 3 x

3 x = \frac{3 x 90}{90}

:حوّل الأعداد لكسور

= 2 6^{\circ} + \frac{3 x \cdot 90}{90}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{234 0 ^{\circ} + 3 x \cdot 90}{90}

3 \cdot 90 = 270 :

اضرب الأعداد

= \frac{234 0 ^{\circ} + 270 x}{90}

= \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )}

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( \frac{270 x + 234 0 ^{\circ}}{90} )}{cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

sin (5 x), cos (\frac{234 0 ^{\circ} + 270 x}{90})

المضاعف المشترك الأصغر لـ:

sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

sin (5 x), cos (\frac{234 0 ^{\circ} + 270 x}{90})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (5 x)

cos (\frac{234 0 ^{\circ} + 270 x}{90})

= sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (5 x) cos (\frac{270 x + 234 0 ^{\circ}}{90})

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( 5 x )}{sin ( 5 x )} :

multiply the denominator and numerator by

cos (\frac{270 x + 234 0 ^{\circ}}{90})

\frac{cos ( 5 x )}{sin ( 5 x )} = \frac{cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

For

\frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )} :

multiply the denominator and numerator by

sin (5 x)

\frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} )} = \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}

= - \frac{cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )} + \frac{sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

= \frac{- cos ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} ) + sin ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )}{sin ( 5 x ) cos ( \frac{270 x + 234 0 ^{\circ}}{90} )}

\frac{- cos ( 5 x ) cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) + sin ( 5 x ) sin ( \frac{234 0 ^{\circ} + 270 x}{90} )}{cos ( \frac{234 0 ^{\circ} + 270 x}{90} ) sin ( 5 x )} = 0

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

- cos (5 x) cos (\frac{234 0 ^{\circ} + 270 x}{90}) + sin (5 x) sin (\frac{234 0 ^{\circ} + 270 x}{90}) = 0

Rewrite using trig identities

- cos (5 x) cos (\frac{234 0 ^{\circ} + 270 x}{90}) + sin (5 x) sin (\frac{234 0 ^{\circ} + 270 x}{90})

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

:فعّل متطابقة الجمع لزوايا

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90})

- cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

- 1

اقسم الطرفين على

- cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

- 1

اقسم الطرفين على

\frac{- cos ( 5 x + \frac{234 0 ^{\circ} + 270 x}{90} )}{- 1} = \frac{0}{- 1}

بسّط

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0

cos (5 x + \frac{234 0 ^{\circ} + 270 x}{90}) = 0 :

حلول عامّة لـ

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n, 5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n, 5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

90

اضرب الطرفين بـ

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 9 0^{\circ} + 36 0^{\circ} n

90

اضرب الطرفين بـ

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

بسّط

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

5 x \cdot 90

بسّط:

450 x

5 x \cdot 90

5 \cdot 90 = 450 :

اضرب الأعداد

= 450 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

بسّط:

234 0^{\circ} + 270 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

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

:اضرب كسور

= \frac{( 234 0 ^{\circ} + 270 x ) \cdot 90}{90}

90 :

إلغ العوامل المشتركة

= 234 0^{\circ} + 270 x

9 0^{\circ} \cdot 90

بسّط:

810 0^{\circ}

9 0^{\circ} \cdot 90

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

:اضرب كسور

= 810 0^{\circ}

\frac{90}{2} = 45 :

اقسم الأعداد

= 810 0^{\circ}

36 0^{\circ} n \cdot 90

بسّط:

3240 0^{\circ} n

36 0^{\circ} n \cdot 90

2 \cdot 90 = 180 :

اضرب الأعداد

= 3240 0^{\circ} n

450 x + 234 0^{\circ} + 270 x = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

انقل

234 0^{\circ}

إلى الجانب الأيمن

720 x + 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

من الطرفين

234 0^{\circ}

اطرح

720 x + 234 0^{\circ} - 234 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n - 234 0^{\circ}

بسّط

720 x = 576 0^{\circ} + 3240 0^{\circ} n

720 x = 576 0^{\circ} + 3240 0^{\circ} n

720

اقسم الطرفين على

720 x = 576 0^{\circ} + 3240 0^{\circ} n

720

اقسم الطرفين على

\frac{720 x}{720} = 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

بسّط

\frac{720 x}{720} = 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

\frac{720 x}{720}

بسّط:

x

\frac{720 x}{720}

\frac{720}{720} = 1 :

اقسم الأعداد

= x

8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

بسّط:

8^{\circ} + \frac{18 0 ^{\circ} n}{4}

8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

8^{\circ}

اختزل:

8^{\circ}

8^{\circ}

16 :

إلغ العوامل المشتركة

= 8^{\circ}

= 8^{\circ} + \frac{3240 0 ^{\circ} n}{720}

\frac{3240 0 ^{\circ} n}{720}

اختزل:

\frac{18 0 ^{\circ} n}{4}

\frac{3240 0 ^{\circ} n}{720}

180 :

إلغ العوامل المشتركة

= \frac{18 0 ^{\circ} n}{4}

= 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

90

اضرب الطرفين بـ

5 x + \frac{234 0 ^{\circ} + 270 x}{90} = 27 0^{\circ} + 36 0^{\circ} n

90

اضرب الطرفين بـ

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

بسّط

5 x \cdot 90 + \frac{234 0 ^{\circ} + 270 x}{90} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

5 x \cdot 90

بسّط:

450 x

5 x \cdot 90

5 \cdot 90 = 450 :

اضرب الأعداد

= 450 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

بسّط:

234 0^{\circ} + 270 x

\frac{234 0 ^{\circ} + 270 x}{90} \cdot 90

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

:اضرب كسور

= \frac{( 234 0 ^{\circ} + 270 x ) \cdot 90}{90}

90 :

إلغ العوامل المشتركة

= 234 0^{\circ} + 270 x

27 0^{\circ} \cdot 90

بسّط:

2430 0^{\circ}

27 0^{\circ} \cdot 90

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

:اضرب كسور

= 2430 0^{\circ}

3 \cdot 90 = 270 :

اضرب الأعداد

= 2430 0^{\circ}

\frac{270}{2} = 135 :

اقسم الأعداد

= 2430 0^{\circ}

36 0^{\circ} n \cdot 90

بسّط:

3240 0^{\circ} n

36 0^{\circ} n \cdot 90

2 \cdot 90 = 180 :

اضرب الأعداد

= 3240 0^{\circ} n

450 x + 234 0^{\circ} + 270 x = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

انقل

234 0^{\circ}

إلى الجانب الأيمن

720 x + 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

من الطرفين

234 0^{\circ}

اطرح

720 x + 234 0^{\circ} - 234 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n - 234 0^{\circ}

بسّط

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

720

اقسم الطرفين على

720 x = 2196 0^{\circ} + 3240 0^{\circ} n

720

اقسم الطرفين على

\frac{720 x}{720} = 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

بسّط

\frac{720 x}{720} = 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

\frac{720 x}{720}

بسّط:

x

\frac{720 x}{720}

\frac{720}{720} = 1 :

اقسم الأعداد

= x

30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

بسّط:

30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

30. 5^{\circ}

اختزل:

30. 5^{\circ}

30. 5^{\circ}

2 :

إلغ العوامل المشتركة

= 30. 5^{\circ}

= 30. 5^{\circ} + \frac{3240 0 ^{\circ} n}{720}

\frac{3240 0 ^{\circ} n}{720}

اختزل:

\frac{18 0 ^{\circ} n}{4}

\frac{3240 0 ^{\circ} n}{720}

180 :

إلغ العوامل المشتركة

= \frac{18 0 ^{\circ} n}{4}

= 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

x = 8^{\circ} + \frac{18 0 ^{\circ} n}{4}, x = 30. 5^{\circ} + \frac{18 0 ^{\circ} n}{4}

رسم

أمثلة شائعة

sin(x)= 3/5 , pi/2 <= x<= pi

sin (x) = \frac{3}{5}, \frac{π}{2} \leq x \leq π

cos(2θ)-sin(2θ)=0

cos (2 θ) - sin (2 θ) = 0

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

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

4sin(x)+3csc(x)=7,0<= x<= 360

4 sin (x) + 3 csc (x) = 7, 0^{\circ} \leq x \leq 36 0^{\circ}

1/(cos(x))=3

\frac{1}{cos ( x )} = 3