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شائع علم المثلثات >

tan(2x+13)=cot(x+53)

الحلّ

tan (2 x + 13) = cot (x + 5 3^{\circ})

الحلّ

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}, x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

راديان

x = \frac{37 π}{540} - \frac{13}{3} + \frac{2 π}{3} n, x = \frac{217 π}{540} - \frac{13}{3} + \frac{2 π}{3} n

خطوات الحلّ

tan (2 x + 13) = cot (x + 5 3^{\circ})

من الطرفين

cot (x + 5 3^{\circ})

اطرح

tan (2 x + 13) - cot (x + 5 3^{\circ}) = 0

sin, cos :

عبّر بواسطة

- cot (5 3^{\circ} + x) + tan (13 + 2 x)

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

:Use the basic trigonometric identity

= - \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + tan (13 + 2 x)

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

:Use the basic trigonometric identity

= - \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + \frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )}

- \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + \frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )}

بسّط:

\frac{- cos ( \frac{954 0 ^{\circ} + 180 x}{180} ) cos ( 2 x + 13 ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} ) cos ( 2 x + 13 )}

- \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + \frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )}

\frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} = \frac{cos ( \frac{954 0 ^{\circ} + 180 x}{180} )}{sin ( \frac{954 0 ^{\circ} + 180 x}{180} )}

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

5 3^{\circ} + x

وحّد:

\frac{954 0 ^{\circ} + 180 x}{180}

5 3^{\circ} + x

x = \frac{x 180}{180}

:حوّل الأعداد لكسور

= 5 3^{\circ} + \frac{x \cdot 180}{180}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{954 0 ^{\circ} + x \cdot 180}{180}

= \frac{cos ( 5 3 ^{\circ} + x )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}

5 3^{\circ} + x

وحّد:

\frac{954 0 ^{\circ} + 180 x}{180}

5 3^{\circ} + x

x = \frac{x 180}{180}

:حوّل الأعداد لكسور

= 5 3^{\circ} + \frac{x \cdot 180}{180}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{954 0 ^{\circ} + x \cdot 180}{180}

= \frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}

= - \frac{cos ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} )} + \frac{sin ( 2 x + 13 )}{cos ( 2 x + 13 )}

sin (\frac{954 0 ^{\circ} + x 180}{180}), cos (13 + 2 x)

المضاعف المشترك الأصغر لـ:

sin (\frac{180 x + 954 0 ^{\circ}}{180}) cos (2 x + 13)

sin (\frac{954 0 ^{\circ} + x \cdot 180}{180}), cos (13 + 2 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (\frac{954 0 ^{\circ} + x 180}{180})

cos (13 + 2 x)

= sin (\frac{180 x + 954 0 ^{\circ}}{180}) cos (2 x + 13)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (\frac{180 x + 954 0 ^{\circ}}{180}) cos (2 x + 13)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )} :

multiply the denominator and numerator by

cos (2 x + 13)

\frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )} = \frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )}

For

\frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )} :

multiply the denominator and numerator by

sin (\frac{180 x + 954 0 ^{\circ}}{180})

\frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )} = \frac{sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{cos ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}

= - \frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )} + \frac{sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{cos ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} ) cos ( 2 x + 13 )}

= \frac{- cos ( \frac{954 0 ^{\circ} + 180 x}{180} ) cos ( 2 x + 13 ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} ) cos ( 2 x + 13 )}

\frac{- cos ( 13 + 2 x ) cos ( \frac{180 x + 954 0 ^{\circ}}{180} ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{cos ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )} = 0

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

- cos (13 + 2 x) cos (\frac{180 x + 954 0 ^{\circ}}{180}) + sin (13 + 2 x) sin (\frac{180 x + 954 0 ^{\circ}}{180}) = 0

Rewrite using trig identities

- cos (13 + 2 x) cos (\frac{180 x + 954 0 ^{\circ}}{180}) + sin (13 + 2 x) sin (\frac{180 x + 954 0 ^{\circ}}{180})

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 (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180})

- cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

- 1

اقسم الطرفين على

- cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

- 1

اقسم الطرفين على

\frac{- cos ( 13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} )}{- 1} = \frac{0}{- 1}

بسّط

cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 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}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n, 13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n, 13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n

انقل

13

إلى الجانب الأيمن

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n

من الطرفين

13

اطرح

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} - 13 = 9 0^{\circ} + 36 0^{\circ} n - 13

بسّط

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n - 13

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n - 13

180

اضرب الطرفين بـ

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n - 13

180

اضرب الطرفين بـ

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 9 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

بسّط

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 9 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

2 x \cdot 180

بسّط:

360 x

2 x \cdot 180

2 \cdot 180 = 360 :

اضرب الأعداد

= 360 x

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180

بسّط:

180 x + 954 0^{\circ}

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180

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

:اضرب كسور

= \frac{( 180 x + 954 0 ^{\circ} ) \cdot 180}{180}

180 :

إلغ العوامل المشتركة

= 180 x + 954 0^{\circ}

9 0^{\circ} \cdot 180

بسّط:

1620 0^{\circ}

9 0^{\circ} \cdot 180

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

:اضرب كسور

= 1620 0^{\circ}

\frac{180}{2} = 90 :

اقسم الأعداد

= 1620 0^{\circ}

36 0^{\circ} n \cdot 180

بسّط:

6480 0^{\circ} n

36 0^{\circ} n \cdot 180

2 \cdot 180 = 360 :

اضرب الأعداد

= 6480 0^{\circ} n

- 13 \cdot 180

بسّط:

- 2340

- 13 \cdot 180

13 \cdot 180 = 2340 :

اضرب الأعداد

= - 2340

360 x + 180 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

انقل

954 0^{\circ}

إلى الجانب الأيمن

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

من الطرفين

954 0^{\circ}

اطرح

540 x + 954 0^{\circ} - 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340 - 954 0^{\circ}

بسّط

540 x = 666 0^{\circ} + 6480 0^{\circ} n - 2340

540 x = 666 0^{\circ} + 6480 0^{\circ} n - 2340

540

اقسم الطرفين على

540 x = 666 0^{\circ} + 6480 0^{\circ} n - 2340

540

اقسم الطرفين على

\frac{540 x}{540} = 12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

بسّط

\frac{540 x}{540} = 12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

\frac{540 x}{540}

بسّط:

x

\frac{540 x}{540}

\frac{540}{540} = 1 :

اقسم الأعداد

= x

12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

بسّط:

12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

جمّع التعابير المتشابهة

= 12.33333 \dots^{\circ} - \frac{2340}{540} + \frac{6480 0 ^{\circ} n}{540}

\frac{2340}{540}

اختزل:

\frac{13}{3}

\frac{2340}{540}

180 :

إلغ العوامل المشتركة

= \frac{13}{3}

= 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{6480 0 ^{\circ} n}{540}

\frac{6480 0 ^{\circ} n}{540}

اختزل:

\frac{36 0 ^{\circ} n}{3}

\frac{6480 0 ^{\circ} n}{540}

180 :

إلغ العوامل المشتركة

= \frac{36 0 ^{\circ} n}{3}

= 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

انقل

13

إلى الجانب الأيمن

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

من الطرفين

13

اطرح

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} - 13 = 27 0^{\circ} + 36 0^{\circ} n - 13

بسّط

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n - 13

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n - 13

180

اضرب الطرفين بـ

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n - 13

180

اضرب الطرفين بـ

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 27 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

بسّط

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 27 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

2 x \cdot 180

بسّط:

360 x

2 x \cdot 180

2 \cdot 180 = 360 :

اضرب الأعداد

= 360 x

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180

بسّط:

180 x + 954 0^{\circ}

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180

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

:اضرب كسور

= \frac{( 180 x + 954 0 ^{\circ} ) \cdot 180}{180}

180 :

إلغ العوامل المشتركة

= 180 x + 954 0^{\circ}

27 0^{\circ} \cdot 180

بسّط:

4860 0^{\circ}

27 0^{\circ} \cdot 180

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

:اضرب كسور

= 4860 0^{\circ}

3 \cdot 180 = 540 :

اضرب الأعداد

= 4860 0^{\circ}

\frac{540}{2} = 270 :

اقسم الأعداد

= 4860 0^{\circ}

36 0^{\circ} n \cdot 180

بسّط:

6480 0^{\circ} n

36 0^{\circ} n \cdot 180

2 \cdot 180 = 360 :

اضرب الأعداد

= 6480 0^{\circ} n

- 13 \cdot 180

بسّط:

- 2340

- 13 \cdot 180

13 \cdot 180 = 2340 :

اضرب الأعداد

= - 2340

360 x + 180 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

انقل

954 0^{\circ}

إلى الجانب الأيمن

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

من الطرفين

954 0^{\circ}

اطرح

540 x + 954 0^{\circ} - 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340 - 954 0^{\circ}

بسّط

540 x = 3906 0^{\circ} + 6480 0^{\circ} n - 2340

540 x = 3906 0^{\circ} + 6480 0^{\circ} n - 2340

540

اقسم الطرفين على

540 x = 3906 0^{\circ} + 6480 0^{\circ} n - 2340

540

اقسم الطرفين على

\frac{540 x}{540} = 72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

بسّط

\frac{540 x}{540} = 72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

\frac{540 x}{540}

بسّط:

x

\frac{540 x}{540}

\frac{540}{540} = 1 :

اقسم الأعداد

= x

72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

بسّط:

72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

جمّع التعابير المتشابهة

= 72.33333 \dots^{\circ} - \frac{2340}{540} + \frac{6480 0 ^{\circ} n}{540}

\frac{2340}{540}

اختزل:

\frac{13}{3}

\frac{2340}{540}

180 :

إلغ العوامل المشتركة

= \frac{13}{3}

= 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{6480 0 ^{\circ} n}{540}

\frac{6480 0 ^{\circ} n}{540}

اختزل:

\frac{36 0 ^{\circ} n}{3}

\frac{6480 0 ^{\circ} n}{540}

180 :

إلغ العوامل المشتركة

= \frac{36 0 ^{\circ} n}{3}

= 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}, x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

رسم

أمثلة شائعة

sin(2x)-sqrt(3)sin(x)=0

sin (2 x) - 3 sin (x) = 0

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

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

-6sqrt(3)cot(θ)+1=-5

- 63 cot (θ) + 1 = - 5

5-10sin(2x)=0

5 - 10 sin (2 x) = 0

solvefor x,cos(3x)=0

so l v e f or x, cos (3 x) = 0