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شائع علم المثلثات >

2sin(x)*tan(x)+5sin(x)=-2cos(x)

الحلّ

2 sin (x) \cdot tan (x) + 5 sin (x) = - 2 cos (x)

الحلّ

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

درجات

x = - 26.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

خطوات الحلّ

2 sin (x) tan (x) + 5 sin (x) = - 2 cos (x)

من الطرفين

- 2 cos (x)

اطرح

2 sin (x) tan (x) + 5 sin (x) + 2 cos (x) = 0

sin, cos :

عبّر بواسطة

2 cos (x) + 5 sin (x) + 2 sin (x) tan (x)

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

:Use the basic trigonometric identity

= 2 cos (x) + 5 sin (x) + 2 sin (x) \frac{sin ( x )}{cos ( x )}

2 cos (x) + 5 sin (x) + 2 sin (x) \frac{sin ( x )}{cos ( x )}

بسّط:

\frac{2 cos ^{2} ( x ) + 5 sin ( x ) cos ( x ) + 2 sin ^{2} ( x )}{cos ( x )}

2 cos (x) + 5 sin (x) + 2 sin (x) \frac{sin ( x )}{cos ( x )}

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

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

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

:اضرب كسور

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

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

sin (x) \cdot 2 sin (x)

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

:فعّل قانون القوى

sin (x) sin (x) = sin^{1 + 1} (x)

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

1 + 1 = 2 :

اجمع الأعداد

= 2 sin^{2} (x)

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

= 2 cos (x) + 5 sin (x) + \frac{2 sin ^{2} ( x )}{cos ( x )}

2 cos (x) = \frac{2 cos ( x ) cos ( x )}{cos ( x )}, 5 sin (x) = \frac{5 sin ( x ) cos ( x )}{cos ( x )}

:حوّل الأعداد لكسور

= \frac{2 cos ( x ) cos ( x )}{cos ( x )} + \frac{5 sin ( x ) cos ( x )}{cos ( x )} + \frac{2 sin ^{2} ( x )}{cos ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{2 cos ( x ) cos ( x ) + 5 sin ( x ) cos ( x ) + 2 sin ^{2} ( x )}{cos ( x )}

2 cos (x) cos (x) + 5 sin (x) cos (x) + 2 sin^{2} (x) = 2 cos^{2} (x) + 5 sin (x) cos (x) + 2 sin^{2} (x)

2 cos (x) cos (x) + 5 sin (x) cos (x) + 2 sin^{2} (x)

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

2 cos (x) cos (x)

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

:فعّل قانون القوى

cos (x) cos (x) = cos^{1 + 1} (x)

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

1 + 1 = 2 :

اجمع الأعداد

= 2 cos^{2} (x)

= 2 cos^{2} (x) + 5 sin (x) cos (x) + 2 sin^{2} (x)

= \frac{2 cos ^{2} ( x ) + 5 sin ( x ) cos ( x ) + 2 sin ^{2} ( x )}{cos ( x )}

= \frac{2 cos ^{2} ( x ) + 5 sin ( x ) cos ( x ) + 2 sin ^{2} ( x )}{cos ( x )}

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x ) + 5 cos ( x ) sin ( x )}{cos ( x )} = 0

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

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x) = 0

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x)

حلل إلى عوامل:

(2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x)

قسّم التعابير لمجموعات

2 sin^{2} (x) + 5 sin (x) cos (x) + 2 cos^{2} (x)

تعريف

Factors of

4 : 1, 2, 4

4

Divisors (Factors)

Find the Prime factors of

4 : 2, 2

4

4 = 2 \cdot 2, 2

ينقسم على

4

= 2 \cdot 2

هو عدد أوّليّ لذلك تحليل آخر لعوامل غير ممكن

2

= 2 \cdot 2

Add the prime factors:

2

Add 1 and the number

4

itself

1, 4

4

قواسم

1, 2, 4

For every two factors such that

u * v = 4,

check if

u + v = 5

Check

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

صحيحCheck

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

خطأ

u = 1, v = 4

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

(2 sin^{2} (x) + sin (x) cos (x)) + (4 sin (x) cos (x) + 2 cos^{2} (x))

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

sin (x) (2 sin (x) + cos (x))

2 sin^{2} (x) + sin (x) cos (x)

من

sin (x)

اخرج العامل

2 sin^{2} (x) + sin (x) cos (x)

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

:فعّل قانون القوى

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

= 2 sin (x) sin (x) + sin (x) cos (x)

sin (x)

قم باخراج العامل المشترك

= sin (x) (2 sin (x) + cos (x))

2 cos (x) (2 sin (x) + cos (x))

4 sin (x) cos (x) + 2 cos^{2} (x)

من

2 cos (x)

اخرج العامل

4 sin (x) cos (x) + 2 cos^{2} (x)

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

:فعّل قانون القوى

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

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

2 \cdot 2

كـ

4

اكتب مجددًا

= 2 \cdot 2 sin (x) cos (x) + 2 cos (x) cos (x)

2 cos (x)

قم باخراج العامل المشترك

= 2 cos (x) (2 sin (x) + cos (x))

= sin (x) (2 sin (x) + cos (x)) + 2 cos (x) (2 sin (x) + cos (x))

2 sin (x) + cos (x)

قم باخراج العامل المشترك

= (2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

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

حلّ كل جزء على حدة

2 sin (x) + cos (x) = 0 or sin (x) + 2 cos (x) = 0

2 sin (x) + cos (x) = 0 : x = arctan (- \frac{1}{2}) + πn

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

Rewrite using trig identities

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

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

اقسم الطرفين على

\frac{2 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{2 sin ( x )}{cos ( x )} + 1 = 0

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

:Use the basic trigonometric identity

2 tan (x) + 1 = 0

2 tan (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

2 tan (x) + 1 = 0

من الطرفين

1

اطرح

2 tan (x) + 1 - 1 = 0 - 1

بسّط

2 tan (x) = - 1

2 tan (x) = - 1

2

اقسم الطرفين على

2 tan (x) = - 1

2

اقسم الطرفين على

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

بسّط

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

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

Apply trig inverse properties

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

tan (x) = - \frac{1}{2} :

حلول عامّة لـ

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

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

Rewrite using trig identities

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

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

اقسم الطرفين على

\frac{sin ( x ) + 2 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{sin ( x )}{cos ( x )} + 2 = 0

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

:Use the basic trigonometric identity

tan (x) + 2 = 0

tan (x) + 2 = 0

انقل

2

إلى الجانب الأيمن

tan (x) + 2 = 0

من الطرفين

2

اطرح

tan (x) + 2 - 2 = 0 - 2

بسّط

tan (x) = - 2

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

tan (x) = - 2 :

حلول عامّة لـ

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 2) + πn

x = arctan (- 2) + πn

وحّد الحلول

x = arctan (- \frac{1}{2}) + πn, x = arctan (- 2) + πn

أظهر الحلّ بالتمثيل العشريّ

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

رسم

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7cos^2(θ)+6sin(θ)-10=-4

7 cos^{2} (θ) + 6 sin (θ) - 10 = - 4

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

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