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شائع علم المثلثات >

sin^4(x)=cos^4(x)

الحلّ

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

الحلّ

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

درجات

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

خطوات الحلّ

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

من الطرفين

cos^{4} (x)

اطرح

sin^{4} (x) - cos^{4} (x) = 0

sin^{4} (x) - cos^{4} (x)

حلل إلى عوامل:

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

sin^{4} (x) - cos^{4} (x)

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

كـ

sin^{4} (x) - cos^{4} (x)

اكتب مجددًا

sin^{4} (x) - cos^{4} (x)

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

:فعّل قانون القوى

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

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

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

:فعّل قانون القوى

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

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

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

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

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

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

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

حلل إلى عوامل:

(sin (x) + cos (x)) (sin (x) - cos (x))

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

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

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

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

= (- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

(- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

بسّط:

(- cos (x) + sin (x)) (cos (x) + sin (x))

(- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

(cos (x) + sin (x)) \cdot 1 = (cos (x) + sin (x)) :

اضرب

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

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

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

حلّ كل جزء على حدة

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

- cos (x) + sin (x) = 0 : x = \frac{π}{4} + πn

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

Rewrite using trig identities

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

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

- 1 + tan (x) = 0

- 1 + tan (x) = 0

انقل

1

إلى الجانب الأيمن

- 1 + tan (x) = 0

للطرفين

1

أضف

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

بسّط

tan (x) = 1

tan (x) = 1

tan (x) = 1 :

حلول عامّة لـ

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) + sin (x) = 0

Rewrite using trig identities

cos (x) + sin (x) = 0

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

1 + tan (x) = 0

1 + tan (x) = 0

انقل

1

إلى الجانب الأيمن

1 + tan (x) = 0

من الطرفين

1

اطرح

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

بسّط

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1 :

حلول عامّة لـ

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

وحّد الحلول

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

رسم

أمثلة شائعة

10.5^2=6.7^2+7.8^2-2*6.7*7.8*cos(x)

10. 5^{2} = 6. 7^{2} + 7. 8^{2} - 2 \cdot 6.7 \cdot 7.8 \cdot cos (x)

3sinh(x)-cosh(x)=1

3 sinh (x) - cosh (x) = 1

tan(45+x/3)=0.58

tan (4 5^{\circ} + \frac{x}{3}) = 0.58

r=a(1+sin(x))

r = a (1 + sin (x))

tan(Θ)=1.4,180<,Θ<270

tan (Θ) = 1.4, 18 0^{\circ} <, Θ < 27 0^{\circ}