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شائع علم المثلثات >

(sin(45)sin(15))/(cos(135)cos(105))

الحلّ

\frac{sin ( 4 5 ^{\circ} ) sin ( 1 5 ^{\circ} )}{cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

الحلّ

1

خطوات الحلّ

\frac{sin ( 4 5 ^{\circ} ) sin ( 1 5 ^{\circ} )}{cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

Rewrite using trig identities:

sin (4 5^{\circ}) sin (1 5^{\circ}) = \frac{cos ( 3 0 ^{\circ} ) - cos ( 6 0 ^{\circ} )}{2}

sin (4 5^{\circ}) sin (1 5^{\circ})

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

:فعّل متطابقة تحويل الضرب لجمع

= \frac{1}{2} (cos (4 5^{\circ} - 1 5^{\circ}) - cos (4 5^{\circ} + 1 5^{\circ}))

بسّط

= \frac{cos ( 3 0 ^{\circ} ) - cos ( 6 0 ^{\circ} )}{2}

= \frac{\frac{c o s ( 3 0 ^{\circ} ) - c o s ( 6 0 ^{\circ} )}{2}}{cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

بسّط

= \frac{cos ( 3 0 ^{\circ} ) - cos ( 6 0 ^{\circ} )}{2 cos ( 13 5 ^{\circ} ) cos ( 10 5 ^{\circ} )}

استخدم المتطابقة الأساسيّة التالية:

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

cos (3 0^{\circ})

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{3}{2}

استخدم المتطابقة الأساسيّة التالية:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

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}

استخدم المتطابقة الأساسيّة التالية:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

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{2}{2}

Rewrite using trig identities:

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

cos (10 5^{\circ})

Rewrite using trig identities:

cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ})

cos (6 0^{\circ} + 4 5^{\circ})

كـ

cos (10 5^{\circ})

أكتب

= cos (6 0^{\circ} + 4 5^{\circ})

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

:فعّل متطابقة الجمع لزوايا

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

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}

استخدم المتطابقة الأساسيّة التالية:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

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{2}{2}

استخدم المتطابقة الأساسيّة التالية:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

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}

استخدم المتطابقة الأساسيّة التالية:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

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{2}{2}

= \frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

بسّط:

\frac{2 ( 1 - 3 )}{4}

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

\frac{2}{2}

قم باخراج العامل المشترك

= \frac{2}{2} (\frac{1}{2} - \frac{3}{2})

\frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2}

\frac{1}{2} - \frac{3}{2}

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

فعّل القانون

= \frac{1 - 3}{2}

= \frac{2}{2} \cdot \frac{1 - 3}{2}

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

:اضرب كسور

= \frac{( 1 - 3 ) 2}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2 ( 1 - 3 )}{4}

= \frac{2 ( 1 - 3 )}{4}

= \frac{\frac{3}{2} - \frac{1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

\frac{\frac{3}{2} - \frac{1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

بسّط:

1

\frac{\frac{3}{2} - \frac{1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

\frac{3}{2} - \frac{1}{2}

وحّد الكسور:

\frac{3 - 1}{2}

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

فعّل القانون

= \frac{3 - 1}{2}

= \frac{\frac{3 - 1}{2}}{2 ( - \frac{2}{2} ) \frac{2 ( 1 - 3 )}{4}}

(- a) = - a

:احذف الأقواس

= \frac{\frac{3 - 1}{2}}{- 2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{\frac{3 - 1}{2}}{2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

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

: استخدم ميزات الكسور التالية

\frac{\frac{3 - 1}{2}}{2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}} = \frac{3 - 1}{2 \cdot 2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

= - \frac{3 - 1}{2 \cdot 2 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

2 \cdot 2 = 4 :

اضرب الأعداد

= - \frac{3 - 1}{4 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}}

4 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}

اضرب بـ:

1 - 3

4 \cdot \frac{2}{2} \cdot \frac{2 ( 1 - 3 )}{4}

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

:اضرب كسور

= \frac{2 2 ( 1 - 3 ) \cdot 4}{2 \cdot 4}

4 :

إلغ العوامل المشتركة

= \frac{2 2 ( 1 - 3 )}{2}

22 (1 - 3) = 2 (1 - 3)

22 (1 - 3)

a a = a

:فعْل قانون الجذور

22 = 2

= 2 (1 - 3)

= \frac{2 ( 1 - 3 )}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= 1 - 3

= - \frac{3 - 1}{1 - 3}

1 - 3 = - (3 - 1)

= \frac{3 - 1}{- ( 3 - 1 )}

بسّط

= - \frac{3 - 1}{3 - 1}

3 - 1 :

إلغ العوامل المشتركة

= - (- 1)

- (- a) = a

فعّل القانون

= 1

= 1

أمثلة شائعة

(sec^2(0)-tan^2(0))/(1+cot^2(0))

\frac{sec ^{2} ( 0 ) - tan ^{2} ( 0 )}{1 + cot ^{2} ( 0 )}

cos(205)

cos (20 5^{\circ})

sin(60)6

sin (6 0^{\circ}) 6

sin^2(15)+cos^2(15)

sin^{2} (1 5^{\circ}) + cos^{2} (1 5^{\circ})

1-cos(180)

1 - cos (18 0^{\circ})