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شائع علم المثلثات >

cos^2(45-a)-sin^2(45-a)=sin^2(a)

الحلّ

cos^{2} (4 5^{\circ} - a) - sin^{2} (4 5^{\circ} - a) = sin^{2} (a)

الحلّ

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

راديان

a = 0 + 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

خطوات الحلّ

cos^{2} (4 5^{\circ} - a) - sin^{2} (4 5^{\circ} - a) = sin^{2} (a)

Rewrite using trig identities

cos^{2} (4 5^{\circ} - a) - sin^{2} (4 5^{\circ} - a) = sin^{2} (a)

Rewrite using trig identities

sin (4 5^{\circ} - a)

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

:فعّل متطابقة الطرح لزوايا

= sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

بسّط:

\frac{2 cos ( a ) - 2 sin ( a )}{2}

sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

sin (4 5^{\circ}) cos (a) = \frac{2 cos ( a )}{2}

sin (4 5^{\circ}) cos (a)

sin (4 5^{\circ})

بسّط:

\frac{2}{2}

sin (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (4 5^{\circ}) = \frac{2}{2}

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}

= \frac{2}{2} cos (a)

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

:اضرب كسور

= \frac{2 cos ( a )}{2}

cos (4 5^{\circ}) sin (a) = \frac{2 sin ( a )}{2}

cos (4 5^{\circ}) sin (a)

cos (4 5^{\circ})

بسّط:

\frac{2}{2}

cos (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (4 5^{\circ}) = \frac{2}{2}

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{2}{2}

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

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

:اضرب كسور

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

= \frac{2 cos ( a )}{2} - \frac{2 sin ( a )}{2}

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

فعّل القانون

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

:فعّل متطابقة الطرح لزوايا

= cos (4 5^{\circ}) cos (a) + sin (4 5^{\circ}) sin (a)

cos (4 5^{\circ}) cos (a) + sin (4 5^{\circ}) sin (a)

بسّط:

\frac{2 cos ( a ) + 2 sin ( a )}{2}

cos (4 5^{\circ}) cos (a) + sin (4 5^{\circ}) sin (a)

cos (4 5^{\circ}) cos (a) = \frac{2 cos ( a )}{2}

cos (4 5^{\circ}) cos (a)

cos (4 5^{\circ})

بسّط:

\frac{2}{2}

cos (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (4 5^{\circ}) = \frac{2}{2}

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{2}{2}

= \frac{2}{2} cos (a)

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

:اضرب كسور

= \frac{2 cos ( a )}{2}

sin (4 5^{\circ}) sin (a) = \frac{2 sin ( a )}{2}

sin (4 5^{\circ}) sin (a)

sin (4 5^{\circ})

بسّط:

\frac{2}{2}

sin (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (4 5^{\circ}) = \frac{2}{2}

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}

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

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

:اضرب كسور

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

= \frac{2 cos ( a )}{2} + \frac{2 sin ( a )}{2}

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

فعّل القانون

= \frac{2 cos ( a ) + 2 sin ( a )}{2}

= \frac{2 cos ( a ) + 2 sin ( a )}{2}

(\frac{2 cos ( a ) + 2 sin ( a )}{2})^{2} - (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2} = sin^{2} (a)

(\frac{2 cos ( a ) + 2 sin ( a )}{2})^{2} - (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

بسّط:

2 cos (a) sin (a)

(\frac{2 cos ( a ) + 2 sin ( a )}{2})^{2} - (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

(\frac{2 cos ( a ) + 2 sin ( a )}{2})^{2} = \frac{( cos ( a ) + sin ( a ) ) ^{2}}{2}

(\frac{2 cos ( a ) + 2 sin ( a )}{2})^{2}

\frac{2 cos ( a ) + 2 sin ( a )}{2} = \frac{cos ( a ) + sin ( a )}{2}

\frac{2 cos ( a ) + 2 sin ( a )}{2}

2

قم باخراج العامل المشترك

= \frac{2 ( cos ( a ) + sin ( a ) )}{2}

\frac{2 ( cos ( a ) + sin ( a ) )}{2}

اختزل:

\frac{cos ( a ) + sin ( a )}{2}

\frac{2 ( cos ( a ) + sin ( a ) )}{2}

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

:فعْل قانون الجذور

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

= \frac{2 ^{\frac{1}{2}} ( cos ( a ) + sin ( a ) )}{2}

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

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

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

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

= \frac{cos ( a ) + sin ( a )}{2 ^{\frac{1}{2}}}

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

:فعْل قانون الجذور

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

= \frac{cos ( a ) + sin ( a )}{2}

= \frac{cos ( a ) + sin ( a )}{2}

= (\frac{cos ( a ) + sin ( a )}{2})^{2}

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

:فعّل قانون القوى

= \frac{( cos ( a ) + sin ( a ) ) ^{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{( cos ( a ) + sin ( a ) ) ^{2}}{2}

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

(\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

\frac{2 cos ( a ) - 2 sin ( a )}{2} = \frac{cos ( a ) - sin ( a )}{2}

\frac{2 cos ( a ) - 2 sin ( a )}{2}

2

قم باخراج العامل المشترك

= \frac{2 ( cos ( a ) - sin ( a ) )}{2}

\frac{2 ( cos ( a ) - sin ( a ) )}{2}

اختزل:

\frac{cos ( a ) - sin ( a )}{2}

\frac{2 ( cos ( a ) - sin ( a ) )}{2}

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

:فعْل قانون الجذور

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

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

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

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

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

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

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

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

:فعْل قانون الجذور

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

= \frac{cos ( a ) - sin ( a )}{2}

= \frac{cos ( a ) - sin ( a )}{2}

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

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

:فعّل قانون القوى

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

= \frac{( cos ( a ) + sin ( a ) ) ^{2}}{2} - \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

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

فعّل القانون

= \frac{( cos ( a ) + sin ( a ) ) ^{2} - ( cos ( a ) - sin ( a ) ) ^{2}}{2}

(cos (a) + sin (a))^{2} - (cos (a) - sin (a))^{2}

وسٌع:

4 cos (a) sin (a)

(cos (a) + sin (a))^{2} - (cos (a) - sin (a))^{2}

(cos (a) + sin (a))^{2} : cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a)

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = cos (a), b = sin (a)

= cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a)

= cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a) - (cos (a) - sin (a))^{2}

(cos (a) - sin (a))^{2} : cos^{2} (a) - 2 cos (a) sin (a) + sin^{2} (a)

(a - b)^{2} = a^{2} - 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = cos (a), b = sin (a)

= cos^{2} (a) - 2 cos (a) sin (a) + sin^{2} (a)

= cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a) - (cos^{2} (a) - 2 cos (a) sin (a) + sin^{2} (a))

- (cos^{2} (a) - 2 cos (a) sin (a) + sin^{2} (a)) : - cos^{2} (a) + 2 cos (a) sin (a) - sin^{2} (a)

- (cos^{2} (a) - 2 cos (a) sin (a) + sin^{2} (a))

افتح أقواس

= - (cos^{2} (a)) - (- 2 cos (a) sin (a)) - (sin^{2} (a))

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

= - cos^{2} (a) + 2 cos (a) sin (a) - sin^{2} (a)

= cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a) - cos^{2} (a) + 2 cos (a) sin (a) - sin^{2} (a)

cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a) - cos^{2} (a) + 2 cos (a) sin (a) - sin^{2} (a)

بسّط:

4 cos (a) sin (a)

cos^{2} (a) + 2 cos (a) sin (a) + sin^{2} (a) - cos^{2} (a) + 2 cos (a) sin (a) - sin^{2} (a)

2 cos (a) sin (a) + 2 cos (a) sin (a) = 4 cos (a) sin (a) :

اجمع العناصر المتشابهة

= cos^{2} (a) + 4 cos (a) sin (a) + sin^{2} (a) - cos^{2} (a) - sin^{2} (a)

cos^{2} (a) - cos^{2} (a) = 0 :

اجمع العناصر المتشابهة

= 4 cos (a) sin (a) + sin^{2} (a) - sin^{2} (a)

sin^{2} (a) - sin^{2} (a) = 0 :

اجمع العناصر المتشابهة

= 4 cos (a) sin (a)

= 4 cos (a) sin (a)

= \frac{4 cos ( a ) sin ( a )}{2}

\frac{4}{2} = 2 :

اقسم الأعداد

= 2 cos (a) sin (a)

2 cos (a) sin (a) = sin^{2} (a)

2 cos (a) sin (a) = sin^{2} (a)

من الطرفين

sin^{2} (a)

اطرح

2 cos (a) sin (a) - sin^{2} (a) = 0

2 cos (a) sin (a) - sin^{2} (a)

حلل إلى عوامل:

sin (a) (2 cos (a) - sin (a))

2 cos (a) sin (a) - sin^{2} (a)

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

sin^{2} (a) = sin (a) sin (a)

= 2 sin (a) cos (a) - sin (a) sin (a)

sin (a)

قم باخراج العامل المشترك

= sin (a) (2 cos (a) - sin (a))

sin (a) (2 cos (a) - sin (a)) = 0

حلّ كل جزء على حدة

sin (a) = 0 or 2 cos (a) - sin (a) = 0

sin (a) = 0 : a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

sin (a) = 0

sin (a) = 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}

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

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

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

حلّ:

a = 36 0^{\circ} n

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

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

a = 36 0^{\circ} n

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

2 cos (a) - sin (a) = 0 : a = arctan (2) + 18 0^{\circ} n

2 cos (a) - sin (a) = 0

Rewrite using trig identities

2 cos (a) - sin (a) = 0

cos (a) \neq = 0, cos (a)

اقسم الطرفين على

\frac{2 cos ( a ) - sin ( a )}{cos ( a )} = \frac{0}{cos ( a )}

بسّط

2 - \frac{sin ( a )}{cos ( a )} = 0

\frac{sin ( x )}{cos ( x )} = tan (x)

:Use the basic trigonometric identity

2 - tan (a) = 0

2 - tan (a) = 0

انقل

2

إلى الجانب الأيمن

2 - tan (a) = 0

من الطرفين

2

اطرح

2 - tan (a) - 2 = 0 - 2

بسّط

- tan (a) = - 2

- tan (a) = - 2

- 1

اقسم الطرفين على

- tan (a) = - 2

- 1

اقسم الطرفين على

\frac{- tan ( a )}{- 1} = \frac{- 2}{- 1}

بسّط

tan (a) = 2

tan (a) = 2

Apply trig inverse properties

tan (a) = 2

tan (a) = 2 :

حلول عامّة لـ

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

a = arctan (2) + 18 0^{\circ} n

a = arctan (2) + 18 0^{\circ} n

وحّد الحلول

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = arctan (2) + 18 0^{\circ} n

أظهر الحلّ بالتمثيل العشريّ

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

رسم

أمثلة شائعة

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

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

cos^2(x)cos(x)=0

cos^{2} (x) cos (x) = 0

2cos(x)=-3

2 cos (x) = - 3

8sec(x)-3tan^2(x)=7

8 sec (x) - 3 tan^{2} (x) = 7

sin^2(6x)+sin^2(3x)=0

sin^{2} (6 x) + sin^{2} (3 x) = 0