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شائع علم المثلثات >

4cos^2(60)+4sin^2(45)-sin^2(30)

الحلّ

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

الحلّ

\frac{11}{4}

عشري

2.75

خطوات الحلّ

4 cos^{2} (6 0^{\circ}) + 4 sin^{2} (4 5^{\circ}) - sin^{2} (3 0^{\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:

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:

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}

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

4 (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2} - (\frac{1}{2})^{2}

بسّط:

\frac{11}{4}

4 (\frac{1}{2})^{2} + 4 (\frac{2}{2})^{2} - (\frac{1}{2})^{2}

جمّع التعابير المتشابهة

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

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

اجمع العناصر المتشابهة

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

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

3 (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2 ^{2}}

(\frac{1}{2})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{1 ^{2}}{2 ^{2}}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{2 ^{2}}

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

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

:اضرب كسور

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

1 \cdot 3 = 3 :

اضرب الأعداد

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

4 (\frac{2}{2})^{2} = 2

4 (\frac{2}{2})^{2}

(\frac{2}{2})^{2} = \frac{1}{2}

(\frac{2}{2})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{( 2 ) ^{2}}{2 ^{2}}

(2)^{2} : 2

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (2^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 2

= \frac{2}{2 ^{2}}

2 :

إلغ العوامل المشتركة

= \frac{1}{2}

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

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

:اضرب كسور

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

1 \cdot 4 = 4 :

اضرب الأعداد

= \frac{4}{2}

\frac{4}{2} = 2 :

اقسم الأعداد

= 2

= \frac{3}{4} + 2

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

2 \cdot 4 + 3 = 11

2 \cdot 4 + 3

2 \cdot 4 = 8 :

اضرب الأعداد

= 8 + 3

8 + 3 = 11 :

اجمع الأعداد

= 11

= \frac{11}{4}

= \frac{11}{4}

أمثلة شائعة

sin((3pi)/(18))

sin (\frac{3 π}{18})

arccos(7/15)

arccos (\frac{7}{15})

tan(-(29pi)/3)

tan (- \frac{29 π}{3})

arcsin(0.11)

arcsin (0.11)

(sin(43.5))/(1.56)

\frac{sin ( 43. 5 ^{\circ} )}{1.56}