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شائع علم المثلثات >

4sin^2(x)+3tan(x)>sec^2(x)

الحلّ

4 sin^{2} (x) + 3 tan (x) > sec^{2} (x)

الحلّ

\frac{π}{12} + πn < x < \frac{5 π}{12} + πn

تدوين الفاصل الزمني

(\frac{π}{12} + πn, \frac{5 π}{12} + πn)

عشري

0.26179 \dots + πn < x < 1.30899 \dots + πn

خطوات الحلّ

4 sin^{2} (x) + 3 tan (x) > sec^{2} (x)

انقل

sec^{2} (x)

إلى الجانب الأيسر

4 sin^{2} (x) + 3 tan (x) > sec^{2} (x)

من الطرفين

sec^{2} (x)

اطرح

4 sin^{2} (x) + 3 tan (x) - sec^{2} (x) > sec^{2} (x) - sec^{2} (x)

4 sin^{2} (x) + 3 tan (x) - sec^{2} (x) > 0

4 sin^{2} (x) + 3 tan (x) - sec^{2} (x) > 0

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

:استخدم المتطابقة التالية

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

لذلك

4 (1 - cos^{2} (x)) + 3 tan (x) - sec^{2} (x) > 0

4 (1 - cos^{2} (x)) + 3 tan (x) - sec^{2} (x)

دوريّة:

π

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

4 (1 - cos^{2} (x)), 3 tan (x), sec^{2} (x)

4 (1 - cos^{2} (x))

دوريّة:

π

زوجيّ

n

مقسمومة على إثنان، إذا تحقّق أنّ

cos (x)

هي دوريّة

cos^{n} (x)

دوريّة

cos (x)

دوريّة:

2 π

2 π

هي

cos (x)

دوريّة

= 2 π

\frac{2 π}{2}

بسّط

π

3 tan (x)

دوريّة:

π

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

دوريّة ال

π

هي

tan (x)

دوريّة

= \frac{π}{∣ 1 ∣}

بسّط

= π

sec^{2} (x)

دوريّة:

π

زوجيّ

n

مقسمومة على إثنان، إذا تحقّق أنّ

sec (x)

هي دوريّة

sec^{n} (x)

دوريّة

sec (x)

دوريّة:

2 π

2 π

هي

sec (x)

دوريّة

= 2 π

\frac{2 π}{2}

بسّط

π

Combine periods:

π, π, π

= π

sin, cos :

عبّر بواسطة

4 (1 - cos^{2} (x)) + 3 tan (x) - sec^{2} (x) > 0

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

:Use the basic trigonometric identity

4 (1 - cos^{2} (x)) + 3 \cdot \frac{sin ( x )}{cos ( x )} - sec^{2} (x) > 0

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

:Use the basic trigonometric identity

4 (1 - cos^{2} (x)) + 3 \cdot \frac{sin ( x )}{cos ( x )} - (\frac{1}{cos ( x )})^{2} > 0

4 (1 - cos^{2} (x)) + 3 \cdot \frac{sin ( x )}{cos ( x )} - (\frac{1}{cos ( x )})^{2} > 0

4 (1 - cos^{2} (x)) + 3 \cdot \frac{sin ( x )}{cos ( x )} - (\frac{1}{cos ( x )})^{2}

بسّط:

\frac{4 cos ^{2} ( x ) ( 1 - cos ^{2} ( x ) ) + 3 sin ( x ) cos ( x ) - 1}{cos ^{2} ( x )}

4 (1 - cos^{2} (x)) + 3 \cdot \frac{sin ( x )}{cos ( x )} - (\frac{1}{cos ( x )})^{2}

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

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

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

:اضرب كسور

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

:حوّل الأعداد لكسور

= \frac{4 ( 1 - cos ^{2} ( x ) )}{1} + \frac{sin ( x ) \cdot 3}{cos ( x )} - \frac{1}{cos ^{2} ( x )}

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

المضاعف المشترك الأصغر لـ:

cos^{2} (x)

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

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= cos^{2} (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos^{2} (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{4 ( 1 - cos ^{2} ( x ) )}{1} :

multiply the denominator and numerator by

cos^{2} (x)

\frac{4 ( 1 - cos ^{2} ( x ) )}{1} = \frac{4 ( 1 - cos ^{2} ( x ) ) cos ^{2} ( x )}{1 \cdot cos ^{2} ( x )} = \frac{4 ( 1 - cos ^{2} ( x ) ) cos ^{2} ( x )}{cos ^{2} ( x )}

For

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

multiply the denominator and numerator by

cos (x)

\frac{sin ( x ) \cdot 3}{cos ( x )} = \frac{sin ( x ) \cdot 3 cos ( x )}{cos ( x ) cos ( x )} = \frac{sin ( x ) \cdot 3 cos ( x )}{cos ^{2} ( x )}

= \frac{4 ( 1 - cos ^{2} ( x ) ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ( x ) \cdot 3 cos ( x )}{cos ^{2} ( x )} - \frac{1}{cos ^{2} ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{4 ( 1 - cos ^{2} ( x ) ) cos ^{2} ( x ) + sin ( x ) \cdot 3 cos ( x ) - 1}{cos ^{2} ( x )}

\frac{4 cos ^{2} ( x ) ( 1 - cos ^{2} ( x ) ) + 3 sin ( x ) cos ( x ) - 1}{cos ^{2} ( x )} > 0

Find the zeroes and undifined points of

\frac{4 cos ^{2} ( x ) ( 1 - cos ^{2} ( x ) ) + 3 sin ( x ) cos ( x ) - 1}{cos ^{2} ( x )}

for

0 \leq x < π

To find the zeroes, set the inequality to zero

\frac{4 cos ^{2} ( x ) ( 1 - cos ^{2} ( x ) ) + 3 sin ( x ) cos ( x ) - 1}{cos ^{2} ( x )} = 0

\frac{4 cos ^{2} ( x ) ( 1 - cos ^{2} ( x ) ) + 3 sin ( x ) cos ( x ) - 1}{cos ^{2} ( x )} = 0, 0 \leq x < π : x = \frac{π}{12}, x = \frac{5 π}{12}

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

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

4 cos^{2} (x) (1 - cos^{2} (x)) + 3 sin (x) cos (x) - 1 = 0

Rewrite using trig identities

- 1 + (1 - cos^{2} (x)) \cdot 4 cos^{2} (x) + 3 cos (x) sin (x)

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

:فعّل نطريّة فيتاغوروس

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

= - 1 + 3 cos (x) sin (x) + 4 cos^{2} (x) sin^{2} (x)

- 1 + 3 cos (x) sin (x) + 4 cos^{2} (x) sin^{2} (x) = 0

- 1 + 3 cos (x) sin (x) + 4 cos^{2} (x) sin^{2} (x)

حلل إلى عوامل:

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

- 1 + 3 cos (x) sin (x) + 4 cos^{2} (x) sin^{2} (x)

قسّم التعابير لمجموعات

4 sin^{2} (x) cos^{2} (x) + 3 sin (x) cos (x) - 1

تعريف

Factors of

4 : 1, 2, 4

4

Divisors (Factors)

Find the Prime factors of

4 : 2, 2

4

4 = 2 \cdot 2, 2

ينقسم على

4

= 2 \cdot 2

هو عدد أوّليّ لذلك تحليل آخر لعوامل غير ممكن

2

= 2 \cdot 2

Add the prime factors:

2

Add 1 and the number

4

itself

1, 4

4

قواسم

1, 2, 4

Negative factors of

4 : - 1, - 2, - 4

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 4

For every two factors such that

u * v = - 4,

check if

u + v = 3

Check

u = 1, v = - 4 : u * v = - 4, u + v = - 3 \Rightarrow

خطأCheck

u = 2, v = - 2 : u * v = - 4, u + v = 0 \Rightarrow

خطأ

u = 4, v = - 1

Group into

(a x^{2} y^{2} + ux y) + (vx y + c)

(4 sin^{2} (x) cos^{2} (x) - sin (x) cos (x)) + (4 sin (x) cos (x) - 1)

= (4 sin^{2} (x) cos^{2} (x) - sin (x) cos (x)) + (4 sin (x) cos (x) - 1)

sin (x) cos (x) (4 sin (x) cos (x) - 1)

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

من

sin (x) cos (x)

اخرج العامل

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

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

:فعّل قانون القوى

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

= 4 sin (x) sin (x) cos (x) cos (x) - sin (x) cos (x)

sin (x) cos (x)

قم باخراج العامل المشترك

= sin (x) cos (x) (4 sin (x) cos (x) - 1)

= sin (x) cos (x) (4 sin (x) cos (x) - 1) + (4 sin (x) cos (x) - 1)

4 sin (x) cos (x) - 1

قم باخراج العامل المشترك

= (4 sin (x) cos (x) - 1) (sin (x) cos (x) + 1)

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

حلّ كل جزء على حدة

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

4 sin (x) cos (x) - 1 = 0, 0 \leq x < π : x = \frac{π}{12}, x = \frac{5 π}{12}

4 sin (x) cos (x) - 1 = 0, 0 \leq x < π

Rewrite using trig identities

4 sin (x) cos (x) - 1

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

:فعّل متطابقة الزاوية المضاعفة

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

= - 1 + 4 \cdot \frac{sin ( 2 x )}{2}

- 1 + 4 \cdot \frac{sin ( 2 x )}{2} = 0

4 \cdot \frac{sin ( 2 x )}{2} = 2 sin (2 x)

4 \cdot \frac{sin ( 2 x )}{2}

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

:اضرب كسور

= \frac{sin ( 2 x ) \cdot 4}{2}

\frac{4}{2} = 2 :

اقسم الأعداد

= 2 sin (2 x)

- 1 + 2 sin (2 x) = 0

انقل

1

إلى الجانب الأيمن

- 1 + 2 sin (2 x) = 0

للطرفين

1

أضف

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

بسّط

2 sin (2 x) = 1

2 sin (2 x) = 1

2

اقسم الطرفين على

2 sin (2 x) = 1

2

اقسم الطرفين على

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

بسّط

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

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

sin (2 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}

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

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

2 x = \frac{π}{6} + 2 πn

حلّ:

x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

2

اقسم الطرفين على

2 x = \frac{π}{6} + 2 πn

2

اقسم الطرفين على

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

بسّط

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

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

بسّط:

\frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{π}{6 \cdot 2}

6 \cdot 2 = 12 :

اضرب الأعداد

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

حلّ:

x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

2

اقسم الطرفين على

2 x = \frac{5 π}{6} + 2 πn

2

اقسم الطرفين على

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

بسّط

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

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

بسّط:

\frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{5 π}{6 \cdot 2}

6 \cdot 2 = 12 :

اضرب الأعداد

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

0 \leq x < π :

حلول للمدى

x = \frac{π}{12}, x = \frac{5 π}{12}

sin (x) cos (x) + 1 = 0, 0 \leq x < π :

لا يوجد حلّ

sin (x) cos (x) + 1 = 0, 0 \leq x < π

Rewrite using trig identities

sin (x) cos (x) + 1

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

:فعّل متطابقة الزاوية المضاعفة

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

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

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

انقل

1

إلى الجانب الأيمن

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

من الطرفين

1

اطرح

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

بسّط

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

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

2

اضرب الطرفين بـ

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

2

اضرب الطرفين بـ

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

بسّط

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

لا يوجد حلّ

وحّد الحلول

x = \frac{π}{12}, x = \frac{5 π}{12}

Find the undefined points:

x = \frac{π}{2}

Find the zeros of the denominator

cos^{2} (x) = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

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}

\frac{π}{12}, \frac{5 π}{12}, \frac{π}{2}

ميّز المقاطع المختلفة

0 < x < \frac{π}{12}, \frac{π}{12} < x < \frac{5 π}{12}, \frac{5 π}{12} < x < \frac{π}{2}, \frac{π}{2} < x < π

لخّص في جدول

4 cos^{2} (x) (1 - cos^{2} (x)) + 3 sin (x) cos (x) - 1 cos^{2} (x) \frac{4 c o s ^{2} ( x ) ( 1 - c o s ^{2} ( x ) ) + 3 s i n ( x ) c o s ( x ) - 1}{c o s ^{2} ( x )} x = 0 - + - 0 < x < \frac{π}{12} - + - x = \frac{π}{12} 0 + 0 \frac{π}{12} < x < \frac{5 π}{12} + + + x = \frac{5 π}{12} 0 + 0 \frac{5 π}{12} < x < \frac{π}{2} - + - x = \frac{π}{2} - 0 غير معرّف \frac{π}{2} < x < π - + - x = π - + -

> 0 :

ميّز المقاطع التي تحقّق الشرط

\frac{π}{12} < x < \frac{5 π}{12}

4 (1 - cos^{2} (x)) + 3 tan (x) - sec^{2} (x) :

استخدم دوريّة الـ

\frac{π}{12} + πn < x < \frac{5 π}{12} + πn

أمثلة شائعة

sin(4x+17)>0

3sin((pix)/(12)-pi/2)<=-2

-sin(x)(2+sin(x))-cos^2(x)>0

1/(sqrt(3))<tan(x)

6cos(θ)>= 0