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شائع علم المثلثات >

(1/2 sin(2x))^2= 1/4 sin^2(4x)

الحلّ

(\frac{1}{2} sin (2 x))^{2} = \frac{1}{4} sin^{2} (4 x)

الحلّ

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

درجات

x = 0^{\circ} + 12 0^{\circ} n, x = 6 0^{\circ} + 12 0^{\circ} n, x = 3 0^{\circ} + 12 0^{\circ} n, x = 9 0^{\circ} + 12 0^{\circ} n

خطوات الحلّ

(\frac{1}{2} sin (2 x))^{2} = \frac{1}{4} sin^{2} (4 x)

من الطرفين

\frac{1}{4} sin^{2} (4 x)

اطرح

\frac{1}{4} sin^{2} (2 x) - \frac{1}{4} sin^{2} (4 x) = 0

\frac{1}{4} sin^{2} (2 x) - \frac{1}{4} sin^{2} (4 x)

بسّط:

\frac{sin ^{2} ( 2 x ) - sin ^{2} ( 4 x )}{4}

\frac{1}{4} sin^{2} (2 x) - \frac{1}{4} sin^{2} (4 x)

\frac{1}{4} sin^{2} (2 x) = \frac{sin ^{2} ( 2 x )}{4}

\frac{1}{4} sin^{2} (2 x)

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

:اضرب كسور

= \frac{1 \cdot sin ^{2} ( 2 x )}{4}

1 \cdot sin^{2} (2 x) = sin^{2} (2 x) :

اضرب

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

\frac{1}{4} sin^{2} (4 x) = \frac{sin ^{2} ( 4 x )}{4}

\frac{1}{4} sin^{2} (4 x)

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

:اضرب كسور

= \frac{1 \cdot sin ^{2} ( 4 x )}{4}

1 \cdot sin^{2} (4 x) = sin^{2} (4 x) :

اضرب

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

= \frac{sin ^{2} ( 2 x )}{4} - \frac{sin ^{2} ( 4 x )}{4}

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

فعّل القانون

= \frac{sin ^{2} ( 2 x ) - sin ^{2} ( 4 x )}{4}

\frac{sin ^{2} ( 2 x ) - sin ^{2} ( 4 x )}{4} = 0

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

sin^{2} (2 x) - sin^{2} (4 x) = 0

sin^{2} (2 x) - sin^{2} (4 x)

حلل إلى عوامل:

(sin (2 x) + sin (4 x)) (sin (2 x) - sin (4 x))

sin^{2} (2 x) - sin^{2} (4 x)

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

sin^{2} (2 x) - sin^{2} (4 x) = (sin (2 x) + sin (4 x)) (sin (2 x) - sin (4 x))

= (sin (2 x) + sin (4 x)) (sin (2 x) - sin (4 x))

(sin (2 x) + sin (4 x)) (sin (2 x) - sin (4 x)) = 0

حلّ كل جزء على حدة

sin (2 x) + sin (4 x) = 0 or sin (2 x) - sin (4 x) = 0

sin (2 x) + sin (4 x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin (2 x) + sin (4 x) = 0

Rewrite using trig identities

sin (2 x) + sin (4 x)

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

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

= 2 sin (\frac{2 x + 4 x}{2}) cos (\frac{2 x - 4 x}{2})

2 sin (\frac{2 x + 4 x}{2}) cos (\frac{2 x - 4 x}{2})

بسّط:

2 cos (x) sin (3 x)

2 sin (\frac{2 x + 4 x}{2}) cos (\frac{2 x - 4 x}{2})

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

\frac{2 x + 4 x}{2}

2 x + 4 x = 6 x :

اجمع العناصر المتشابهة

= \frac{6 x}{2}

\frac{6}{2} = 3 :

اقسم الأعداد

= 3 x

= 2 sin (3 x) cos (\frac{2 x - 4 x}{2})

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

\frac{2 x - 4 x}{2}

2 x - 4 x = - 2 x :

اجمع العناصر المتشابهة

= \frac{- 2 x}{2}

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

: استخدم ميزات الكسور التالية

= - \frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= - x

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

cos (- x) = cos (x)

:Use the negative angle identity

= 2 cos (x) sin (3 x)

= 2 cos (x) sin (3 x)

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

حلّ كل جزء على حدة

cos (x) = 0 or sin (3 x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0 :

حلول عامّة لـ

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin (3 x) = 0

sin (3 x) = 0 :

حلول عامّة لـ

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

3 x = 0 + 2 πn, 3 x = π + 2 πn

3 x = 0 + 2 πn, 3 x = π + 2 πn

3 x = 0 + 2 πn

حلّ:

x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

3

اقسم الطرفين على

3 x = 2 πn

3

اقسم الطرفين على

\frac{3 x}{3} = \frac{2 πn}{3}

بسّط

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

3 x = π + 2 πn

حلّ:

x = \frac{π}{3} + \frac{2 πn}{3}

3 x = π + 2 πn

3

اقسم الطرفين على

3 x = π + 2 πn

3

اقسم الطرفين على

\frac{3 x}{3} = \frac{π}{3} + \frac{2 πn}{3}

بسّط

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin (2 x) - sin (4 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

sin (2 x) - sin (4 x) = 0

Rewrite using trig identities

sin (2 x) - sin (4 x)

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

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

= 2 sin (\frac{2 x - 4 x}{2}) cos (\frac{2 x + 4 x}{2})

2 sin (\frac{2 x - 4 x}{2}) cos (\frac{2 x + 4 x}{2})

بسّط:

- 2 cos (3 x) sin (x)

2 sin (\frac{2 x - 4 x}{2}) cos (\frac{2 x + 4 x}{2})

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

\frac{2 x - 4 x}{2}

2 x - 4 x = - 2 x :

اجمع العناصر المتشابهة

= \frac{- 2 x}{2}

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

: استخدم ميزات الكسور التالية

= - \frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= - x

= 2 sin (- x) cos (\frac{2 x + 4 x}{2})

sin (- x) = - sin (x)

:Use the negative angle identity

= 2 cos (\frac{2 x + 4 x}{2}) (- sin (x))

(- a) = - a

:احذف الأقواس

= - 2 cos (\frac{2 x + 4 x}{2}) sin (x)

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

\frac{2 x + 4 x}{2}

2 x + 4 x = 6 x :

اجمع العناصر المتشابهة

= \frac{6 x}{2}

\frac{6}{2} = 3 :

اقسم الأعداد

= 3 x

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

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

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

حلّ كل جزء على حدة

cos (3 x) = 0 or sin (x) = 0

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

cos (3 x) = 0

cos (3 x) = 0 :

حلول عامّة لـ

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

3 x = \frac{π}{2} + 2 πn, 3 x = \frac{3 π}{2} + 2 πn

3 x = \frac{π}{2} + 2 πn, 3 x = \frac{3 π}{2} + 2 πn

3 x = \frac{π}{2} + 2 πn

حلّ:

x = \frac{π}{6} + \frac{2 πn}{3}

3 x = \frac{π}{2} + 2 πn

3

اقسم الطرفين على

3 x = \frac{π}{2} + 2 πn

3

اقسم الطرفين على

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

بسّط

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

\frac{3 x}{3}

بسّط:

x

\frac{3 x}{3}

\frac{3}{3} = 1 :

اقسم الأعداد

= x

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

بسّط:

\frac{π}{6} + \frac{2 πn}{3}

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

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

: استخدم ميزات الكسور التالية

= \frac{π}{2 \cdot 3}

2 \cdot 3 = 6 :

اضرب الأعداد

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

3 x = \frac{3 π}{2} + 2 πn

حلّ:

x = \frac{π}{2} + \frac{2 πn}{3}

3 x = \frac{3 π}{2} + 2 πn

3

اقسم الطرفين على

3 x = \frac{3 π}{2} + 2 πn

3

اقسم الطرفين على

\frac{3 x}{3} = \frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

بسّط

\frac{3 x}{3} = \frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

\frac{3 x}{3}

بسّط:

x

\frac{3 x}{3}

\frac{3}{3} = 1 :

اقسم الأعداد

= x

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

بسّط:

\frac{π}{2} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

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

: استخدم ميزات الكسور التالية

= \frac{3 π}{2 \cdot 3}

2 \cdot 3 = 6 :

اضرب الأعداد

= \frac{3 π}{6}

3 :

إلغ العوامل المشتركة

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0 :

حلول عامّة لـ

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

وحّد الحلول

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

ادمج المجالات المتطابقة

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

رسم

أمثلة شائعة

8sin^2(x)-3sin(x)-2=0

8 sin^{2} (x) - 3 sin (x) - 2 = 0

6sin(x)+12=15

6 sin (x) + 12 = 15

cot^2(x)cos(x)=cos(x)

cot^{2} (x) cos (x) = cos (x)

1+2cos(x)+cos^2(x)=3sin^2(x)

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

solvefor L,sin(kL)=0

so l v e f or L, sin (k L) = 0