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شائع علم المثلثات >

sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y))

الحلّ

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

الحلّ

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

خطوات الحلّ

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

بالاستعانة بطريقة التعويض

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

cos (x) = u :

على افتراض أنّ

1 - u^{2} = 0.75 1 - cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.75 1 - cos^{2} (y)

ربّع الطرفين:

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y)

(1 - u^{2})^{2} = (0.75 1 - cos^{2} (y))^{2}

(1 - u^{2})^{2}

وسّع:

1 - u^{2}

(1 - u^{2})^{2}

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

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

= (1 - u^{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

= 1 - u^{2}

(0.75 1 - cos^{2} (y))^{2}

وسّع:

0.5625 - 0.5625 cos^{2} (y)

(0.75 1 - cos^{2} (y))^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 0.7 5^{2} (- cos^{2} (y) + 1)^{2}

(1 - cos^{2} (y))^{2} : 1 - cos^{2} (y)

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

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

= (1 - cos^{2} (y))^{\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

= 1 - cos^{2} (y)

= 0.7 5^{2} (1 - cos^{2} (y))

0.7 5^{2} = 0.5625

= 0.5625 (- cos^{2} (y) + 1)

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 0.5625, b = 1, c = cos^{2} (y)

= 0.5625 \cdot 1 - 0.5625 cos^{2} (y)

= 1 \cdot 0.5625 - 0.5625 cos^{2} (y)

1 \cdot 0.5625 = 0.5625 :

اضرب الأعداد

= 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

حلّ:

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

انقل

1

إلى الجانب الأيمن

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

من الطرفين

1

اطرح

1 - u^{2} - 1 = 0.5625 - 0.5625 cos^{2} (y) - 1

بسّط

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

- 1

اقسم الطرفين على

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

- 1

اقسم الطرفين على

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

بسّط

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

\frac{- u ^{2}}{- 1}

بسّط:

u^{2}

\frac{- u ^{2}}{- 1}

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= u^{2}

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

بسّط:

0.5625 cos^{2} (y) + 0.4375

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

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

فعّل القانون

= \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{- 1}

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

: استخدم ميزات الكسور التالية

= - \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{1}

\frac{a}{1} = a

فعّل القانون

= - (- 0.5625 cos^{2} (y) - 0.4375)

افتح أقواس

= - (- 0.5625 cos^{2} (y)) - (- 0.4375)

فعّل قوانين سالب-موجب

- (- a) = a

= 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

افحص الإجبات:

u = 0.5625 cos^{2} (y) + 0.4375

صحيح

, u = - 0.5625 cos^{2} (y) + 0.4375

صحيح

للتحقّق من دقّة الحلول

1 - u^{2} = 0.75 1 - cos^{2} (y)

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

u = 0.5625 cos^{2} (y) + 0.4375

استبدل:

صحيح

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

1 - (0.5625 cos^{2} (y) + 0.4375)^{2}

بسّط:

- 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)^{2}

(0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(0.5625 cos^{2} (y) + 0.4375)^{2}

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

:فعْل قانون الجذور

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

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

:فعّل قانون القوى

= (0.5625 cos^{2} (y) + 0.4375)^{\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

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

1 - (0.5625 cos^{2} (y) + 0.4375)

وسٌع:

- 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

افتح أقواس

= - (0.5625 cos^{2} (y)) - (0.4375)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

1 - 0.4375 = 0.5625 :

اطرح الأعداد

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

صحيح

u = - 0.5625 cos^{2} (y) + 0.4375

استبدل:

صحيح

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2}

بسّط:

- 0.5625 cos^{2} (y) + 0.5625

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2}

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(- 0.5625 cos^{2} (y) + 0.4375)^{2}

زوجيّ n

إذا تحقّق أنّ

, (- a)^{n} = a^{n}

:فعّل قانون القوى

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = (0.5625 cos^{2} (y) + 0.4375)^{2}

= (0.5625 cos^{2} (y) + 0.4375)^{2}

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

:فعْل قانون الجذور

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

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

:فعّل قانون القوى

= (0.5625 cos^{2} (y) + 0.4375)^{\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

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

1 - (0.5625 cos^{2} (y) + 0.4375)

وسٌع:

- 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

افتح أقواس

= - (0.5625 cos^{2} (y)) - (0.4375)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

1 - 0.4375 = 0.5625 :

اطرح الأعداد

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

صحيح

The solutions are

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

u = cos (x)

استبدل مجددًا

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375 : x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = 0.5625 cos^{2} (y) + 0.4375

Apply trig inverse properties

cos (x) = 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375 :

حلول عامّة لـ

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375 : x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

Apply trig inverse properties

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = - 0.5625 cos^{2} (y) + 0.4375 :

حلول عامّة لـ

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

وحّد الحلول

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

رسم

أمثلة شائعة

sec^2(x)-6tan(x)+4=0

sec^{2} (x) - 6 tan (x) + 4 = 0

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

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

-5tan(x)-12=tan(x)-8

- 5 tan (x) - 12 = tan (x) - 8

5=sec^2(x)+3

5 = sec^{2} (x) + 3

sin(θ)= 24/25

sin (θ) = \frac{24}{25}