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شائع علم المثلثات >

50/33 =(sin(x))/(sin(120-x))

الحلّ

\frac{50}{33} = \frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )}

الحلّ

x = 1.38810 \dots + 18 0^{\circ} n

راديان

x = 1.38810 \dots + πn

خطوات الحلّ

\frac{50}{33} = \frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )}

بدّل الأطراف

\frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )} = \frac{50}{33}

Rewrite using trig identities

\frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )} = \frac{50}{33}

Rewrite using trig identities

sin (12 0^{\circ} - x)

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

:فعّل متطابقة الطرح لزوايا

= sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

بسّط:

\frac{3}{2} cos (x) + \frac{1}{2} sin (x)

sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

sin (12 0^{\circ})

بسّط:

\frac{3}{2}

sin (12 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (12 0^{\circ}) = \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{3}{2} cos (x) - cos (12 0^{\circ}) sin (x)

cos (12 0^{\circ})

بسّط:

- \frac{1}{2}

cos (12 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (12 0^{\circ}) = - \frac{1}{2}

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}

= - \frac{1}{2}

= \frac{3}{2} cos (x) - (- \frac{1}{2} sin (x))

- (- a) = a

فعّل القانون

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{50}{33}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{50}{33}

من الطرفين

\frac{50}{33}

اطرح

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33} = 0

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33}

بسّط:

\frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{2 sin ( x )}{3 cos ( x ) + sin ( x )}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}

\frac{3}{2} cos (x) = \frac{3 cos ( x )}{2}

\frac{3}{2} cos (x)

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

:اضرب كسور

= \frac{3 cos ( x )}{2}

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

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

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

:اضرب كسور

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

1 \cdot sin (x) = sin (x) :

اضرب

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

= \frac{sin ( x )}{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}

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

وحّد الكسور:

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

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

فعّل القانون

= \frac{3 cos ( x ) + sin ( x )}{2}

= \frac{sin ( x )}{\frac{3 c o s ( x ) + s i n ( x )}{2}}

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

: استخدم ميزات الكسور التالية

= \frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )}

= \frac{2 sin ( x )}{3 cos ( x ) + sin ( x )} - \frac{50}{33}

3 cos (x) + sin (x), 33

المضاعف المشترك الأصغر لـ:

33 (3 cos (x) + sin (x))

3 cos (x) + sin (x), 33

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

3 cos (x) + sin (x)

33

= 33 (3 cos (x) + sin (x))

اكتب مجددًا الكسور بحيث يكون المقام مشترك

33 (3 cos (x) + sin (x))

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )} :

multiply the denominator and numerator by

33

\frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )} = \frac{sin ( x ) \cdot 2 \cdot 33}{( 3 cos ( x ) + sin ( x ) ) \cdot 33} = \frac{66 sin ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

For

\frac{50}{33} :

multiply the denominator and numerator by

3 cos (x) + sin (x)

\frac{50}{33} = \frac{50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

= \frac{66 sin ( x )}{33 ( 3 cos ( x ) + sin ( x ) )} - \frac{50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{66 sin ( x ) - 50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

66 sin (x) - 50 (3 cos (x) + sin (x))

وسٌع:

16 sin (x) - 503 cos (x)

66 sin (x) - 50 (3 cos (x) + sin (x))

- 50 (3 cos (x) + sin (x))

وسٌع:

- 503 cos (x) - 50 sin (x)

- 50 (3 cos (x) + sin (x))

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = - 50, b = 3 cos (x), c = sin (x)

= - 503 cos (x) + (- 50) sin (x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 503 cos (x) - 50 sin (x)

= 66 sin (x) - 503 cos (x) - 50 sin (x)

66 sin (x) - 50 sin (x) = 16 sin (x) :

اجمع العناصر المتشابهة

= 16 sin (x) - 503 cos (x)

= \frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

\frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )} = 0

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

16 sin (x) - 503 cos (x) = 0

Rewrite using trig identities

16 sin (x) - 503 cos (x) = 0

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

اقسم الطرفين على

\frac{16 sin ( x ) - 50 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{16 sin ( x )}{cos ( x )} - 503 = 0

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

:Use the basic trigonometric identity

16 tan (x) - 503 = 0

16 tan (x) - 503 = 0

انقل

503

إلى الجانب الأيمن

16 tan (x) - 503 = 0

للطرفين

503

أضف

16 tan (x) - 503 + 503 = 0 + 503

بسّط

16 tan (x) = 503

16 tan (x) = 503

16

اقسم الطرفين على

16 tan (x) = 503

16

اقسم الطرفين على

\frac{16 tan ( x )}{16} = \frac{50 3}{16}

بسّط

tan (x) = \frac{25 3}{8}

tan (x) = \frac{25 3}{8}

Apply trig inverse properties

tan (x) = \frac{25 3}{8}

tan (x) = \frac{25 3}{8} :

حلول عامّة لـ

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{25 3}{8}) + 18 0^{\circ} n

x = arctan (\frac{25 3}{8}) + 18 0^{\circ} n

أظهر الحلّ بالتمثيل العشريّ

x = 1.38810 \dots + 18 0^{\circ} n

رسم

أمثلة شائعة

cot(x)csc(x)=cos(x)

cot (x) csc (x) = cos (x)

2tan(x)=sqrt(2)

2 tan (x) = 2

0.5=cos(pi/6 x)

0.5 = cos (\frac{π}{6} x)

cos^2(x)= pi/4

cos^{2} (x) = \frac{π}{4}

cot^2(θ)+csc(θ)=1,0<= θ<2pi

cot^{2} (θ) + csc (θ) = 1, 0 \leq θ < 2 π