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شائع علم المثلثات >

sin(2x)=tan(x)

الحلّ

sin (2 x) = tan (x)

الحلّ

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

خطوات الحلّ

sin (2 x) = tan (x)

من الطرفين

tan (x)

اطرح

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

sin, cos :

عبّر بواسطة

sin (2 x) - tan (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

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

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

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

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

:فعّل قانون القوى

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

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

1 + 1 = 2 :

اجمع الأعداد

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

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

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

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

حلل إلى عوامل:

sin (x) (2 cos (x) + 1) (2 cos (x) - 1)

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

sin (x)

قم باخراج العامل المشترك

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

2 cos^{2} (x) - 1

حلل إلى عوامل:

(2 cos (x) + 1) (2 cos (x) - 1)

2 cos^{2} (x) - 1

(2 cos (x))^{2} - 1^{2}

كـ

2 cos^{2} (x) - 1

اكتب مجددًا

2 cos^{2} (x) - 1

a = (a)^{2}

:فعْل قانون الجذور

2 = (2)^{2}

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

1^{2}

كـ

1

اكتب مجددًا

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

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

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

= (2 cos (x) + 1) (2 cos (x) - 1)

= sin (x) (2 cos (x) + 1) (2 cos (x) - 1)

sin (x) (2 cos (x) + 1) (2 cos (x) - 1) = 0

حلّ كل جزء على حدة

sin (x) = 0 or 2 cos (x) + 1 = 0 or 2 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

2 cos (x) + 1 = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 cos (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

2 cos (x) + 1 = 0

من الطرفين

1

اطرح

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

بسّط

2 cos (x) = - 1

2 cos (x) = - 1

2

اقسم الطرفين على

2 cos (x) = - 1

2

اقسم الطرفين على

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

بسّط

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

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

بسّط:

cos (x)

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

2 :

إلغ العوامل المشتركة

= cos (x)

\frac{- 1}{2}

بسّط:

- \frac{2}{2}

\frac{- 1}{2}

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{1}{2}

- \frac{1}{2}

حوّل لصيغة عدد كسريّ:

- \frac{2}{2}

- \frac{1}{2}

\frac{2}{2}

اضرب بالمرافق

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

1 \cdot 2 = 2

22 = 2

22

a a = a

:فعْل قانون الجذور

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (x) = - \frac{2}{2}

cos (x) = - \frac{2}{2}

cos (x) = - \frac{2}{2}

cos (x) = - \frac{2}{2} :

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}

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 cos (x) - 1 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 cos (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

2 cos (x) - 1 = 0

للطرفين

1

أضف

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

بسّط

2 cos (x) = 1

2 cos (x) = 1

2

اقسم الطرفين على

2 cos (x) = 1

2

اقسم الطرفين على

\frac{2 cos ( x )}{2} = \frac{1}{2}

بسّط

\frac{2 cos ( x )}{2} = \frac{1}{2}

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

بسّط:

cos (x)

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

2 :

إلغ العوامل المشتركة

= cos (x)

\frac{1}{2}

بسّط:

\frac{2}{2}

\frac{1}{2}

\frac{2}{2}

اضرب بالمرافق

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

a a = a

:فعْل قانون الجذور

22 = 2

= 2

= \frac{2}{2}

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

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

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

cos (x) = \frac{2}{2} :

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}

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

وحّد الحلول

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

رسم

أمثلة شائعة

cos(x)=0.35

cos (x) = 0.35

sin((17pi)/6)

sin (\frac{17 π}{6})

cos(2θ)= 1/2

cos (2 θ) = \frac{1}{2}

tan(arcsin(-4/5))

tan (arcsin (- \frac{4}{5}))

sin^2(pi/3)

sin^{2} (\frac{π}{3})