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شائع علم المثلثات >

cos(x)-cos(2x)=0,0<= x<360

الحلّ

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

الحلّ

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

راديان

x = 0, x = \frac{2 π}{3}, x = \frac{4 π}{3}

خطوات الحلّ

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

Rewrite using trig identities

- cos (2 x) + cos (x)

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

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

بسّط:

2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

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

x + 2 x = 3 x :

اجمع العناصر المتشابهة

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

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

\frac{x - 2 x}{2}

x - 2 x = - x :

اجمع العناصر المتشابهة

= \frac{- x}{2}

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

: استخدم ميزات الكسور التالية

= - \frac{x}{2}

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

sin (- x) = - sin (x)

:Use the negative angle identity

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

- (- a) = a

فعّل القانون

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

حلّ كل جزء على حدة

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0, 0 \leq x < 36 0^{\circ} : x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

sin (\frac{3 x}{2}) = 0, 0 \leq x < 36 0^{\circ}

sin (\frac{3 x}{2}) = 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}

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} = 0 + 36 0^{\circ} n

حلّ:

x = \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 0 + 36 0^{\circ} n

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

\frac{3 x}{2} = 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{3 x}{2} = 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n

بسّط

3 x = 72 0^{\circ} n

3 x = 72 0^{\circ} n

3

اقسم الطرفين على

3 x = 72 0^{\circ} n

3

اقسم الطرفين على

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3}

بسّط

x = \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

بسّط

3 x = 36 0^{\circ} + 72 0^{\circ} n

3 x = 36 0^{\circ} + 72 0^{\circ} n

3

اقسم الطرفين على

3 x = 36 0^{\circ} + 72 0^{\circ} n

3

اقسم الطرفين على

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

بسّط

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

0 \leq x < 36 0^{\circ} :

حلول للمدى

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

sin (\frac{x}{2}) = 0, 0 \leq x < 36 0^{\circ} : x = 0

sin (\frac{x}{2}) = 0, 0 \leq x < 36 0^{\circ}

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

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n

حلّ:

x = 72 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n

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

\frac{x}{2} = 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{x}{2} = 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{2 x}{2} = 2 \cdot 36 0^{\circ} n

بسّط

x = 72 0^{\circ} n

x = 72 0^{\circ} n

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 36 0^{\circ} + 72 0^{\circ} n

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

2

اضرب الطرفين بـ

\frac{2 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

بسّط

x = 36 0^{\circ} + 72 0^{\circ} n

x = 36 0^{\circ} + 72 0^{\circ} n

x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

0 \leq x < 36 0^{\circ} :

حلول للمدى

x = 0

وحّد الحلول

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

رسم

أمثلة شائعة

3=-3cos(θ)

3 = - 3 cos (θ)

2cos(θ)-sqrt(3)=0,0<= θ<= 2pi

2 cos (θ) - 3 = 0, 0 \leq θ \leq 2 π

-4sin(x)=cos^2(x)+1,0<= x<= 2pi

- 4 sin (x) = cos^{2} (x) + 1, 0 \leq x \leq 2 π

tan(2x)cos(2x)+cot(2x)sin(2x)=1

tan (2 x) cos (2 x) + cot (2 x) sin (2 x) = 1

sin^2(2x)-cos^2(2x)= 1/2

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