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tan(x)>= sin(2x)

الحلّ

tan (x) \geq sin (2 x)

الحلّ

\frac{π}{4} + πn \leq x < \frac{π}{2} + πn or \frac{3 π}{4} + πn \leq x \leq π + πn

تدوين الفاصل الزمني

[\frac{π}{4} + πn, \frac{π}{2} + πn) \cup [\frac{3 π}{4} + πn, π + πn]

العشري

0.78539 \dots + πn \leq x < 1.57079 \dots + πn or 2.35619 \dots + πn \leq x \leq 3.14159 \dots + πn

خطوات الحلّ

tan (x) \geq sin (2 x)

انقل

sin (2 x)

إلى الجانب الأيسر

tan (x) \geq sin (2 x)

من الطرفين

sin (2 x)

اطرح

tan (x) - sin (2 x) \geq sin (2 x) - sin (2 x)

tan (x) - sin (2 x) \geq 0

tan (x) - sin (2 x) \geq 0

tan (x) - sin (2 x)

دوريّة:

π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

tan (x), sin (2 x)

tan (x)

دوريّة:

π

π

هي

tan (x)

دوريّة

= π

sin (2 x)

دوريّة:

π

a sin (b x \pm c) \pm d = \frac{sin دوريّة}{∣ b ∣}

دوريّة ال

2 π

هي

sin (x)

دوريّة

= \frac{2 π}{∣ 2 ∣}

بسّط

= π

Combine periods:

π, π

= π

sin, cos :

عبّر بواسطة

tan (x) - sin (2 x) \geq 0

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

:Use the basic trigonometric identity

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

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

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

Find the zeroes and undifined points of

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

for

0 \leq x < π

To find the zeroes, set the inequality to zero

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

\frac{sin ( x ) - sin ( 2 x ) cos ( x )}{cos ( x )} = 0, 0 \leq x < π : x = 0, x = \frac{3 π}{4}, x = \frac{π}{4}

\frac{sin ( x ) - sin ( 2 x ) cos ( x )}{cos ( x )} = 0, 0 \leq x < π

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

sin (x) - sin (2 x) cos (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, 0 \leq x < π : x = 0

sin (x) = 0, 0 \leq x < π

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

0 \leq x < π :

حلول للمدى

x = 0

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

2 cos (x) + 1 = 0, 0 \leq x < π

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

0 \leq x < π :

حلول للمدى

x = \frac{3 π}{4}

2 cos (x) - 1 = 0, 0 \leq x < π : x = \frac{π}{4}

2 cos (x) - 1 = 0, 0 \leq x < π

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

0 \leq x < π :

حلول للمدى

x = \frac{π}{4}

وحّد الحلول

x = 0, x = \frac{3 π}{4}, x = \frac{π}{4}

Find the undefined points:

x = \frac{π}{2}

Find the zeros of the denominator

cos (x) = 0

cos (x) = 0 :

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

0 \leq x < π :

حلول للمدى

x = \frac{π}{2}

0, \frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}

ميّز المقاطع المختلفة

0 < x < \frac{π}{4}, \frac{π}{4} < x < \frac{π}{2}, \frac{π}{2} < x < \frac{3 π}{4}, \frac{3 π}{4} < x < π

لخّص في جدول

sin (x) - sin (2 x) cos (x) cos (x) \frac{s i n ( x ) - s i n ( 2 x ) c o s ( x )}{c o s ( x )} x = 0 0 + 0 0 < x < \frac{π}{4} - + - x = \frac{π}{4} 0 + 0 \frac{π}{4} < x < \frac{π}{2} + + + x = \frac{π}{2} + 0 غير معرّف \frac{π}{2} < x < \frac{3 π}{4} + - - x = \frac{3 π}{4} 0 - 0 \frac{3 π}{4} < x < π - - + x = π 0 - 0

\geq 0 :

ميّز المقاطع التي تحقّق الشرط

x = 0 or x = \frac{π}{4} or \frac{π}{4} < x < \frac{π}{2} or x = \frac{3 π}{4} or \frac{3 π}{4} < x < π or x = π

ادمج المجالات المتطابقة

x = 0 or \frac{π}{4} \leq x < \frac{π}{2} or \frac{3 π}{4} \leq x < π or x = π