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شائع علم المثلثات >

sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0

الحلّ

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) = 0

الحلّ

x = 1.37340 \dots + πn, x = 1.47112 \dots + πn

درجات

x = 78.69006 \dots^{\circ} + 18 0^{\circ} n, x = 84.28940 \dots^{\circ} + 18 0^{\circ} n

خطوات الحلّ

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) = 0

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

حلل إلى عوامل:

(sin (x) - 5 cos (x)) (sin (x) - 10 cos (x))

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

قسّم التعابير لمجموعات

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

تعريف

Factors of

50 : 1, 2, 5, 10, 25, 50

50

Divisors (Factors)

Find the Prime factors of

50 : 2, 5, 5

50

50 = 25 \cdot 2, 2

ينقسم على

50

= 2 \cdot 25

25 = 5 \cdot 5, 5

ينقسم على

25

= 2 \cdot 5 \cdot 5

مركّب من أعداد أوّليّة فقط، لذلك تحليل إضافيّ غير ممكن

2, 5

= 2 \cdot 5 \cdot 5

Multiply the prime factors of

50 : 10, 25

2 \cdot 5 = 10

5 \cdot 5 = 25

10, 25

10, 25

Add the prime factors:

2, 5

Add 1 and the number

50

itself

1, 50

50

قواسم

1, 2, 5, 10, 25, 50

Negative factors of

50 : - 1, - 2, - 5, - 10, - 25, - 50

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 5, - 10, - 25, - 50

For every two factors such that

u * v = 50,

check if

u + v = - 15

Check

u = 1, v = 50 : u * v = 50, u + v = 51 \Rightarrow

خطأCheck

u = 2, v = 25 : u * v = 50, u + v = 27 \Rightarrow

خطأ

u = - 5, v = - 10

Group into

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

(sin^{2} (x) - 5 sin (x) cos (x)) + (- 10 sin (x) cos (x) + 50 cos^{2} (x))

= (sin^{2} (x) - 5 sin (x) cos (x)) + (- 10 sin (x) cos (x) + 50 cos^{2} (x))

sin (x) (sin (x) - 5 cos (x))

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

من

sin (x)

اخرج العامل

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

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

:فعّل قانون القوى

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

= sin (x) sin (x) - 5 sin (x) cos (x)

sin (x)

قم باخراج العامل المشترك

= sin (x) (sin (x) - 5 cos (x))

- 10 cos (x) (sin (x) - 5 cos (x))

- 10 sin (x) cos (x) + 50 cos^{2} (x)

من

- 10 cos (x)

اخرج العامل

- 10 sin (x) cos (x) + 50 cos^{2} (x)

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

:فعّل قانون القوى

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

= - 10 sin (x) cos (x) + 50 cos (x) cos (x)

5 \cdot 10

كـ

50

اكتب مجددًا

= - 10 sin (x) cos (x) + 5 \cdot 10 cos (x) cos (x)

- 10 cos (x)

قم باخراج العامل المشترك

= - 10 cos (x) (sin (x) - 5 cos (x))

= sin (x) (sin (x) - 5 cos (x)) - 10 cos (x) (sin (x) - 5 cos (x))

sin (x) - 5 cos (x)

قم باخراج العامل المشترك

= (sin (x) - 5 cos (x)) (sin (x) - 10 cos (x))

(sin (x) - 5 cos (x)) (sin (x) - 10 cos (x)) = 0

حلّ كل جزء على حدة

sin (x) - 5 cos (x) = 0 or sin (x) - 10 cos (x) = 0

sin (x) - 5 cos (x) = 0 : x = arctan (5) + πn

sin (x) - 5 cos (x) = 0

Rewrite using trig identities

sin (x) - 5 cos (x) = 0

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

اقسم الطرفين على

\frac{sin ( x ) - 5 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{sin ( x )}{cos ( x )} - 5 = 0

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

:Use the basic trigonometric identity

tan (x) - 5 = 0

tan (x) - 5 = 0

انقل

5

إلى الجانب الأيمن

tan (x) - 5 = 0

للطرفين

5

أضف

tan (x) - 5 + 5 = 0 + 5

بسّط

tan (x) = 5

tan (x) = 5

Apply trig inverse properties

tan (x) = 5

tan (x) = 5 :

حلول عامّة لـ

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

x = arctan (5) + πn

x = arctan (5) + πn

sin (x) - 10 cos (x) = 0 : x = arctan (10) + πn

sin (x) - 10 cos (x) = 0

Rewrite using trig identities

sin (x) - 10 cos (x) = 0

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

اقسم الطرفين على

\frac{sin ( x ) - 10 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{sin ( x )}{cos ( x )} - 10 = 0

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

:Use the basic trigonometric identity

tan (x) - 10 = 0

tan (x) - 10 = 0

انقل

10

إلى الجانب الأيمن

tan (x) - 10 = 0

للطرفين

10

أضف

tan (x) - 10 + 10 = 0 + 10

بسّط

tan (x) = 10

tan (x) = 10

Apply trig inverse properties

tan (x) = 10

tan (x) = 10 :

حلول عامّة لـ

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

x = arctan (10) + πn

x = arctan (10) + πn

وحّد الحلول

x = arctan (5) + πn, x = arctan (10) + πn

أظهر الحلّ بالتمثيل العشريّ

x = 1.37340 \dots + πn, x = 1.47112 \dots + πn

رسم

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