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شائع علم المثلثات >

cos(4θ)=cos(2θ),180<= θ<= 360

الحلّ

cos (4 θ) = cos (2 θ), 18 0^{\circ} \leq θ \leq 36 0^{\circ}

الحلّ

θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

+1

راديان

θ = π, θ = \frac{4 π}{3}, θ = \frac{5 π}{3}, θ = 2 π

خطوات الحلّ

cos (4 θ) = cos (2 θ), 18 0^{\circ} \leq θ \leq 36 0^{\circ}

من الطرفين

cos (2 θ)

اطرح

cos (4 θ) - cos (2 θ) = 0

Rewrite using trig identities

- cos (2 θ) + cos (4 θ)

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

:

Eq u a t i o n 0 :

متطابقة تحويل الجمع لضرب

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

- 2 sin (\frac{4 θ + 2 θ}{2}) sin (\frac{4 θ - 2 θ}{2})

بسّط:

- 2 sin (3 θ) sin (θ)

- 2 sin (\frac{4 θ + 2 θ}{2}) sin (\frac{4 θ - 2 θ}{2})

\frac{4 θ + 2 θ}{2} = 3 θ

\frac{4 θ + 2 θ}{2}

4 θ + 2 θ = 6 θ :

اجمع العناصر المتشابهة

= \frac{6 θ}{2}

\frac{6}{2} = 3 :

اقسم الأعداد

= 3 θ

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

\frac{4 θ - 2 θ}{2} = θ

\frac{4 θ - 2 θ}{2}

4 θ - 2 θ = 2 θ :

اجمع العناصر المتشابهة

= \frac{2 θ}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= θ

= - 2 sin (3 θ) sin (θ)

= - 2 sin (3 θ) sin (θ)

- 2 sin (3 θ) sin (θ) = 0

حلّ كل جزء على حدة

sin (3 θ) = 0 or sin (θ) = 0

sin (3 θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ} : θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

sin (3 θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ}

sin (3 θ) = 0 :

حلول عامّة لـ

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}

3 θ = 0 + 36 0^{\circ} n, 3 θ = 18 0^{\circ} + 36 0^{\circ} n

3 θ = 0 + 36 0^{\circ} n, 3 θ = 18 0^{\circ} + 36 0^{\circ} n

3 θ = 0 + 36 0^{\circ} n

حلّ:

θ = \frac{36 0 ^{\circ} n}{3}

3 θ = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

3 θ = 36 0^{\circ} n

3

اقسم الطرفين على

3 θ = 36 0^{\circ} n

3

اقسم الطرفين على

\frac{3 θ}{3} = \frac{36 0 ^{\circ} n}{3}

بسّط

θ = \frac{36 0 ^{\circ} n}{3}

θ = \frac{36 0 ^{\circ} n}{3}

3 θ = 18 0^{\circ} + 36 0^{\circ} n

حلّ:

θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

3 θ = 18 0^{\circ} + 36 0^{\circ} n

3

اقسم الطرفين على

3 θ = 18 0^{\circ} + 36 0^{\circ} n

3

اقسم الطرفين على

\frac{3 θ}{3} = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

بسّط

θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = \frac{36 0 ^{\circ} n}{3}, θ = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

18 0^{\circ} \leq θ \leq 36 0^{\circ} :

حلول للمدى

θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

sin (θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ} : θ = 18 0^{\circ}, θ = 36 0^{\circ}

sin (θ) = 0, 18 0^{\circ} \leq θ \leq 36 0^{\circ}

sin (θ) = 0 :

حلول عامّة لـ

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}

θ = 0 + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n

حلّ:

θ = 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

θ = 36 0^{\circ} n

θ = 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

18 0^{\circ} \leq θ \leq 36 0^{\circ} :

حلول للمدى

θ = 18 0^{\circ}, θ = 36 0^{\circ}

وحّد الحلول

θ = 18 0^{\circ}, θ = 24 0^{\circ}, θ = 30 0^{\circ}, θ = 36 0^{\circ}

رسم

أمثلة شائعة

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8 cos (2 x) = 8 - 8 cos (2 x)

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- (2) cos (10 t) = (6) sin (10 t)

cot (x) = \frac{1}{4}

tan(x)=0,0<x<2pi

tan (x) = 0, 0 < x < 2 π