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شائع علم المثلثات >

(2sin(x)-1)*(sqrt(3)tan(x)+1)>0

الحلّ

(2 sin (x) - 1) \cdot (3 tan (x) + 1) > 0

الحلّ

\frac{π}{6} + 2 πn < x < \frac{π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < \frac{11 π}{6} + 2 πn

تدوين الفاصل الزمني

(\frac{π}{6} + 2 πn, \frac{π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, \frac{11 π}{6} + 2 πn)

عشري

0.52359 \dots + 2 πn < x < 1.57079 \dots + 2 πn or 4.71238 \dots + 2 πn < x < 5.75958 \dots + 2 πn

خطوات الحلّ

(2 sin (x) - 1) (3 tan (x) + 1) > 0

(2 sin (x) - 1) (3 tan (x) + 1)

دوريّة:

2 π

:

مركّبة من الدوالّ والدوريّات التالية

(2 sin (x) - 1) (3 tan (x) + 1)

2 π

مع دوريّات

sin (x)

:

الدوريّة المركّبة للدالّة هي

= 2 π

sin, cos :

عبّر بواسطة

(2 sin (x) - 1) (3 tan (x) + 1) > 0

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

:Use the basic trigonometric identity

(2 sin (x) - 1) (3 \frac{sin ( x )}{cos ( x )} + 1) > 0

(2 sin (x) - 1) (3 \frac{sin ( x )}{cos ( x )} + 1) > 0

(2 sin (x) - 1) (3 \frac{sin ( x )}{cos ( x )} + 1)

بسّط:

\frac{( 3 sin ( x ) + cos ( x ) ) ( 2 sin ( x ) - 1 )}{cos ( x )}

(2 sin (x) - 1) (3 \frac{sin ( x )}{cos ( x )} + 1)

3 \frac{sin ( x )}{cos ( x )}

اضرب بـ:

\frac{3 sin ( x )}{cos ( x )}

3 \frac{sin ( x )}{cos ( x )}

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

:اضرب كسور

= \frac{sin ( x ) 3}{cos ( x )}

= (2 sin (x) - 1) (\frac{3 sin ( x )}{cos ( x )} + 1)

\frac{sin ( x ) 3}{cos ( x )} + 1

وحّد:

\frac{3 sin ( x ) + cos ( x )}{cos ( x )}

\frac{sin ( x ) 3}{cos ( x )} + 1

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

:حوّل الأعداد لكسور

= \frac{sin ( x ) 3}{cos ( x )} + \frac{1 \cdot 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 ) 3 + 1 \cdot cos ( x )}{cos ( x )}

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

اضرب

= \frac{3 sin ( x ) + cos ( x )}{cos ( x )}

= \frac{3 sin ( x ) + cos ( x )}{cos ( x )} (2 sin (x) - 1)

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

:اضرب كسور

= \frac{( sin ( x ) 3 + cos ( x ) ) ( 2 sin ( x ) - 1 )}{cos ( x )}

\frac{( 3 sin ( x ) + cos ( x ) ) ( 2 sin ( x ) - 1 )}{cos ( x )} > 0

Find the zeroes and undifined points of

\frac{( 3 sin ( x ) + cos ( x ) ) ( 2 sin ( x ) - 1 )}{cos ( x )}

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

\frac{( 3 sin ( x ) + cos ( x ) ) ( 2 sin ( x ) - 1 )}{cos ( x )} = 0

\frac{( 3 sin ( x ) + cos ( x ) ) ( 2 sin ( x ) - 1 )}{cos ( x )} = 0, 0 \leq x < 2 π : x = \frac{5 π}{6}, x = \frac{11 π}{6}, x = \frac{π}{6}

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

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

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

حلّ كل جزء على حدة

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

3 sin (x) + cos (x) = 0, 0 \leq x < 2 π : x = \frac{5 π}{6}, x = \frac{11 π}{6}

3 sin (x) + cos (x) = 0, 0 \leq x < 2 π

Rewrite using trig identities

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

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

اقسم الطرفين على

\frac{3 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{3 sin ( x )}{cos ( x )} + 1 = 0

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

:Use the basic trigonometric identity

3 tan (x) + 1 = 0

3 tan (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

3 tan (x) + 1 = 0

من الطرفين

1

اطرح

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

بسّط

3 tan (x) = - 1

3 tan (x) = - 1

3

اقسم الطرفين على

3 tan (x) = - 1

3

اقسم الطرفين على

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

بسّط

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

\frac{3 tan ( x )}{3}

بسّط:

tan (x)

\frac{3 tan ( x )}{3}

3 :

إلغ العوامل المشتركة

= tan (x)

\frac{- 1}{3}

بسّط:

- \frac{3}{3}

\frac{- 1}{3}

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

: استخدم ميزات الكسور التالية

= - \frac{1}{3}

- \frac{1}{3}

حوّل لصيغة عدد كسريّ:

- \frac{3}{3}

- \frac{1}{3}

\frac{3}{3}

اضرب بالمرافق

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3} :

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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

0 \leq x < 2 π :

حلول للمدى

x = \frac{5 π}{6}, x = \frac{11 π}{6}

2 sin (x) - 1 = 0, 0 \leq x < 2 π : x = \frac{π}{6}, x = \frac{5 π}{6}

2 sin (x) - 1 = 0, 0 \leq x < 2 π

انقل

1

إلى الجانب الأيمن

2 sin (x) - 1 = 0

للطرفين

1

أضف

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

بسّط

2 sin (x) = 1

2 sin (x) = 1

2

اقسم الطرفين على

2 sin (x) = 1

2

اقسم الطرفين على

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

بسّط

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

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

sin (x) = \frac{1}{2} :

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 = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

0 \leq x < 2 π :

حلول للمدى

x = \frac{π}{6}, x = \frac{5 π}{6}

وحّد الحلول

x = \frac{5 π}{6}, x = \frac{11 π}{6}, x = \frac{π}{6}

Find the undefined points:

x = \frac{π}{2}, x = \frac{3 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 < 2 π :

حلول للمدى

x = \frac{π}{2}, x = \frac{3 π}{2}

\frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}, \frac{3 π}{2}, \frac{11 π}{6}

ميّز المقاطع المختلفة

0 < x < \frac{π}{6}, \frac{π}{6} < x < \frac{π}{2}, \frac{π}{2} < x < \frac{5 π}{6}, \frac{5 π}{6} < x < \frac{3 π}{2}, \frac{3 π}{2} < x < \frac{11 π}{6}, \frac{11 π}{6} < x < 2 π

لخّص في جدول

3 sin (x) + cos (x) 2 sin (x) - 1 cos (x) \frac{( 3 s i n ( x ) + c o s ( x ) ) ( 2 s i n ( x ) - 1 )}{c o s ( x )} x = 0 + - + - 0 < x < \frac{π}{6} + - + - x = \frac{π}{6} + 0 + 0 \frac{π}{6} < x < \frac{π}{2} + + + + x = \frac{π}{2} + + 0 غير معرّف \frac{π}{2} < x < \frac{5 π}{6} + + - - x = \frac{5 π}{6} 00 - 0 \frac{5 π}{6} < x < \frac{3 π}{2} - - - - x = \frac{3 π}{2} - - 0 غير معرّف \frac{3 π}{2} < x < \frac{11 π}{6} - - + + x = \frac{11 π}{6} 0 - + 0 \frac{11 π}{6} < x < 2 π + - + - x = 2 π + - + -

> 0 :

ميّز المقاطع التي تحقّق الشرط

\frac{π}{6} < x < \frac{π}{2} or \frac{3 π}{2} < x < \frac{11 π}{6}

(2 sin (x) - 1) (3 tan (x) + 1) :

استخدم دوريّة الـ

\frac{π}{6} + 2 πn < x < \frac{π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < \frac{11 π}{6} + 2 πn

أمثلة شائعة

(2cos(x)-1)(2cos(x)+sqrt(2))<0

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

2cos(3x-1/2)>= (sqrt(2))/2

2 cos (3 x - \frac{1}{2}) \geq \frac{2}{2}

2cos(x)+sqrt(2)<0

2 cos (x) + 2 < 0

sin(2*x)>= 1

sin (2 \cdot x) \geq 1

-5cos(x)>0

- 5 cos (x) > 0