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شائع علم المثلثات >

(sin(105)-sin(15))/(sin(105)+sin(15))

الحلّ

\frac{sin ( 10 5 ^{\circ} ) - sin ( 1 5 ^{\circ} )}{sin ( 10 5 ^{\circ} ) + sin ( 1 5 ^{\circ} )}

الحلّ

\frac{3}{3}

عشري

0.57735 \dots

خطوات الحلّ

\frac{sin ( 10 5 ^{\circ} ) - sin ( 1 5 ^{\circ} )}{sin ( 10 5 ^{\circ} ) + sin ( 1 5 ^{\circ} )}

Rewrite using trig identities:

sin (10 5^{\circ}) = \frac{2 ( 3 + 1 )}{4}

sin (10 5^{\circ})

Rewrite using trig identities:

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

sin (10 5^{\circ})

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

كـ

sin (10 5^{\circ})

أكتب

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

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

:فعّل متطابقة الجمع لزوايا

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

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

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:

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:

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:

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

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

بسّط:

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

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

\frac{2}{2}

قم باخراج العامل المشترك

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

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

\frac{3}{2} + \frac{1}{2}

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

فعّل القانون

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

2 \cdot 2 = 4 :

اضرب الأعداد

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

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

Rewrite using trig identities:

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

sin (1 5^{\circ})

Rewrite using trig identities:

sin (4 5^{\circ}) cos (3 0^{\circ}) - cos (4 5^{\circ}) sin (3 0^{\circ})

sin (1 5^{\circ})

sin (4 5^{\circ} - 3 0^{\circ})

كـ

sin (1 5^{\circ})

أكتب

= sin (4 5^{\circ} - 3 0^{\circ})

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

:فعّل متطابقة الطرح لزوايا

= sin (4 5^{\circ}) cos (3 0^{\circ}) - cos (4 5^{\circ}) sin (3 0^{\circ})

= sin (4 5^{\circ}) cos (3 0^{\circ}) - cos (4 5^{\circ}) sin (3 0^{\circ})

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}

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 (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 (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

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

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

بسّط:

\frac{6 - 2}{4}

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

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

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

:اضرب كسور

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

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2 3}{4}

23

بسّط:

6

23

a b = a \cdot b

:فعْل قانون الجذور

23 = 2 \cdot 3

= 2 \cdot 3

2 \cdot 3 = 6 :

اضرب الأعداد

= 6

= \frac{6}{4}

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

\frac{2}{2} \cdot \frac{1}{2}

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

:اضرب كسور

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

2 \cdot 1 = 2 :

اضرب

= \frac{2}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2}{4}

= \frac{6}{4} - \frac{2}{4}

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

فعّل القانون

= \frac{6 - 2}{4}

= \frac{6 - 2}{4}

= \frac{\frac{2 ( 3 + 1 )}{4} - \frac{6 - 2}{4}}{\frac{2 ( 3 + 1 )}{4} + \frac{6 - 2}{4}}

\frac{\frac{2 ( 3 + 1 )}{4} - \frac{6 - 2}{4}}{\frac{2 ( 3 + 1 )}{4} + \frac{6 - 2}{4}}

بسّط:

\frac{3}{3}

\frac{\frac{2 ( 3 + 1 )}{4} - \frac{6 - 2}{4}}{\frac{2 ( 3 + 1 )}{4} + \frac{6 - 2}{4}}

\frac{2 ( 1 + 3 )}{4} + \frac{6 - 2}{4}

وحّد الكسور:

\frac{2 ( 3 + 1 ) + 6 - 2}{4}

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

فعّل القانون

= \frac{2 ( 1 + 3 ) + 6 - 2}{4}

= \frac{\frac{2 ( 1 + 3 )}{4} - \frac{6 - 2}{4}}{\frac{2 ( 1 + 3 ) + 6 - 2}{4}}

\frac{2 ( 1 + 3 )}{4} - \frac{6 - 2}{4}

وحّد الكسور:

\frac{2 ( 3 + 1 ) - ( 6 - 2 )}{4}

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

فعّل القانون

= \frac{2 ( 1 + 3 ) - ( 6 - 2 )}{4}

= \frac{\frac{2 ( 1 + 3 ) - ( 6 - 2 )}{4}}{\frac{2 ( 1 + 3 ) + 6 - 2}{4}}

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

:اقسم الكسور

= \frac{( 2 ( 3 + 1 ) - ( 6 - 2 ) ) \cdot 4}{4 ( 2 ( 3 + 1 ) + 6 - 2 )}

4 :

إلغ العوامل المشتركة

= \frac{2 ( 3 + 1 ) - ( 6 - 2 )}{2 ( 3 + 1 ) + 6 - 2}

2 (3 + 1) + 6 - 2

وسٌع:

26

2 (3 + 1) + 6 - 2

2 (3 + 1)

وسٌع:

6 + 2

2 (3 + 1)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = 2, b = 3, c = 1

= 23 + 2 \cdot 1

= 23 + 1 \cdot 2

23 + 1 \cdot 2

بسّط:

6 + 2

23 + 1 \cdot 2

23 = 6

23

a b = a \cdot b

:فعْل قانون الجذور

23 = 2 \cdot 3

= 2 \cdot 3

2 \cdot 3 = 6 :

اضرب الأعداد

= 6

1 \cdot 2 = 2

1 \cdot 2

1 \cdot 2 = 2 :

اضرب

= 2

= 6 + 2

= 6 + 2

= 6 + 2 + 6 - 2

6 + 2 + 6 - 2

بسّط:

26

6 + 2 + 6 - 2

2 - 2 = 0 :

اجمع العناصر المتشابهة

= 6 + 6

6 + 6 = 26 :

اجمع العناصر المتشابهة

= 26

= 26

= \frac{2 ( 1 + 3 ) - ( 6 - 2 )}{2 6}

2 (3 + 1) - (6 - 2)

وسٌع:

22

2 (3 + 1) - (6 - 2)

2 (3 + 1)

وسٌع:

6 + 2

2 (3 + 1)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = 2, b = 3, c = 1

= 23 + 2 \cdot 1

= 23 + 1 \cdot 2

23 + 1 \cdot 2

بسّط:

6 + 2

23 + 1 \cdot 2

23 = 6

23

a b = a \cdot b

:فعْل قانون الجذور

23 = 2 \cdot 3

= 2 \cdot 3

2 \cdot 3 = 6 :

اضرب الأعداد

= 6

1 \cdot 2 = 2

1 \cdot 2

1 \cdot 2 = 2 :

اضرب

= 2

= 6 + 2

= 6 + 2

= 6 + 2 - (6 - 2)

- (6 - 2) : - 6 + 2

- (6 - 2)

افتح أقواس

= - (6) - (- 2)

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

= - 6 + 2

= 6 + 2 - 6 + 2

6 + 2 - 6 + 2

بسّط:

22

6 + 2 - 6 + 2

2 + 2 = 22 :

اجمع العناصر المتشابهة

= 6 + 22 - 6

6 - 6 = 0 :

اجمع العناصر المتشابهة

= 22

= 22

= \frac{2 2}{2 6}

\frac{2}{2} = 1 :

اقسم الأعداد

= \frac{2}{6}

6

حلل إلى عوامل:

23

6 = 2 \cdot 3

حلّل إلى عوامل

= 2 \cdot 3

n ab = n a n b

:فعْل قانون الجذور

= 23

= \frac{2}{2 3}

2 :

إلغ العوامل المشتركة

= \frac{1}{3}

\frac{1}{3}

حوّل لصيغة عدد كسريّ:

\frac{3}{3}

\frac{1}{3}

\frac{3}{3}

اضرب بالمرافق

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

1 \cdot 3 = 3

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

= \frac{3}{3}

أمثلة شائعة

-5.2sin(3.71797)-8cos(3.71797)

- 5.2 sin (3.71797) - 8 cos (3.71797)

sin(arcsin(-4))

sin (arcsin (- 4))

sin(2*1)

sin (2 \cdot 1)

cos(pi/9)cos(pi/9)

cos (\frac{π}{9}) cos (\frac{π}{9})

500*cos(30)

500 \cdot cos (3 0^{\circ})