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شائع علم المثلثات >

(1-cos(a))(1+cos(a))=tan(a)sin(a)

الحلّ

(1 - cos (a)) (1 + cos (a)) = tan (a) sin (a)

الحلّ

a = 2 πn, a = π + 2 πn

درجات

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

خطوات الحلّ

(1 - cos (a)) (1 + cos (a)) = tan (a) sin (a)

من الطرفين

tan (a) sin (a)

اطرح

(1 - cos (a)) (1 + cos (a)) - tan (a) sin (a) = 0

Rewrite using trig identities

(1 + cos (a)) (1 - cos (a)) - sin (a) tan (a)

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

1 - cos (a)

\frac{1 + cos ( a )}{1 + cos ( a )}

اضرب بـ

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

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

فعّل قانون فرق المربّعات

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

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

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

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

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

:اضرب كسور

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

1 + cos (a) :

إلغ العوامل المشتركة

= sin^{2} (a)

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

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

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

حلل إلى عوامل:

sin (a) (sin (a) - tan (a))

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

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

:فعّل قانون القوى

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

= sin (a) sin (a) - sin (a) tan (a)

sin (a)

قم باخراج العامل المشترك

= sin (a) (sin (a) - tan (a))

sin (a) (sin (a) - tan (a)) = 0

حلّ كل جزء على حدة

sin (a) = 0 or sin (a) - tan (a) = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

sin (a) = 0 :

حلول عامّة لـ

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn

حلّ:

a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

sin (a) - tan (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) - tan (a) = 0

sin, cos :

عبّر بواسطة

sin (a) - tan (a)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

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

- sin (a) + cos (a) sin (a)

حلل إلى عوامل:

sin (a) (cos (a) - 1)

- sin (a) + cos (a) sin (a)

sin (a)

قم باخراج العامل المشترك

= sin (a) (- 1 + cos (a))

sin (a) (cos (a) - 1) = 0

حلّ كل جزء على حدة

sin (a) = 0 or cos (a) - 1 = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

sin (a) = 0 :

حلول عامّة لـ

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn

حلّ:

a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

انقل

1

إلى الجانب الأيمن

cos (a) - 1 = 0

للطرفين

1

أضف

cos (a) - 1 + 1 = 0 + 1

بسّط

cos (a) = 1

cos (a) = 1

cos (a) = 1 :

حلول عامّة لـ

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = 0 + 2 πn

a = 0 + 2 πn

a = 0 + 2 πn

حلّ:

a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

وحّد الحلول

a = 2 πn, a = π + 2 πn

وحّد الحلول

a = 2 πn, a = π + 2 πn

رسم

أمثلة شائعة

tan^2(x)+1/6+(tan(1))/3 =0

tan^{2} (x) + \frac{1}{6} + \frac{tan ( 1 )}{3} = 0

-2cos^2(x)-5sin(x)+5=0

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

3-4sin^3(x)=sin^3(x)

3 - 4 sin^{3} (x) = sin^{3} (x)

cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x)

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

sin(2x)=5cos(x)

sin (2 x) = 5 cos (x)