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شائع علم المثلثات >

tan(t)-tan^2(t)+sec^3(t)>0

الحلّ

tan (t) - tan^{2} (t) + sec^{3} (t) > 0

الحلّ

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

تدوين الفاصل الزمني

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

عشري

2 πn \leq t < 1.57079 \dots + 2 πn or 4.71238 \dots + 2 πn < t \leq 6.28318 \dots + 2 πn

خطوات الحلّ

tan (t) - tan^{2} (t) + sec^{3} (t) > 0

tan (t) - tan^{2} (t) + sec^{3} (t)

دوريّة:

2 π

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

tan (t), tan^{2} (t), sec^{3} (t)

tan (t)

دوريّة:

π

π

هي

tan (x)

دوريّة

= π

tan^{2} (t)

دوريّة:

π

P er i o d i c i t y o f tan^{n} (x) =

Periodicity of

tan (x)

tan (t)

دوريّة:

π

π

هي

tan (x)

دوريّة

= π

π

sec^{3} (t)

دوريّة:

2 π

Periodicity of

sec^{n} (x) =

Periodicity of

sec (x),

if n is odd

sec (t)

دوريّة:

2 π

2 π

هي

sec (x)

دوريّة

= 2 π

2 π

Combine periods:

π, π, 2 π

= 2 π

sin, cos :

عبّر بواسطة

tan (t) - tan^{2} (t) + sec^{3} (t) > 0

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

:Use the basic trigonometric identity

\frac{sin ( t )}{cos ( t )} - (\frac{sin ( t )}{cos ( t )})^{2} + sec^{3} (t) > 0

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

:Use the basic trigonometric identity

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

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

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

بسّط:

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

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

(\frac{sin ( t )}{cos ( t )})^{2} = \frac{sin ^{2} ( t )}{cos ^{2} ( t )}

(\frac{sin ( t )}{cos ( t )})^{2}

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

:فعّل قانون القوى

= \frac{sin ^{2} ( t )}{cos ^{2} ( t )}

(\frac{1}{cos ( t )})^{3} = \frac{1}{cos ^{3} ( t )}

(\frac{1}{cos ( t )})^{3}

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

:فعّل قانون القوى

= \frac{1 ^{3}}{cos ^{3} ( t )}

1^{a} = 1

فعّل القانون

1^{3} = 1

= \frac{1}{cos ^{3} ( t )}

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

cos (t), cos^{2} (t), cos^{3} (t)

المضاعف المشترك الأصغر لـ:

cos^{3} (t)

cos (t), cos^{2} (t), cos^{3} (t)

Lowest Common Multiplier (LCM)

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

= cos^{3} (t)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos^{3} (t)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{sin ( t )}{cos ( t )} :

multiply the denominator and numerator by

cos^{2} (t)

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

For

\frac{sin ^{2} ( t )}{cos ^{2} ( t )} :

multiply the denominator and numerator by

cos (t)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

Find the zeroes and undifined points of

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

for

0 \leq t < 2 π

To find the zeroes, set the inequality to zero

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

Find the undefined points:

t = \frac{π}{2}, t = \frac{3 π}{2}

Find the zeros of the denominator

cos^{3} (t) = 0

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

فعّل القانون

cos (t) = 0

cos (t) = 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}

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

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

0 \leq t < 2 π :

حلول للمدى

t = \frac{π}{2}, t = \frac{3 π}{2}

\frac{π}{2}, \frac{3 π}{2}

ميّز المقاطع المختلفة

0 < t < \frac{π}{2}, \frac{π}{2} < t < \frac{3 π}{2}, \frac{3 π}{2} < t < 2 π

لخّص في جدول

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

> 0 :

ميّز المقاطع التي تحقّق الشرط

t = 0 or 0 < t < \frac{π}{2} or \frac{3 π}{2} < t < 2 π or t = 2 π

ادمج المجالات المتطابقة

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

اتحاد مجالين هو مجموعة الأرقام الموجودة بواحد أو أكثر من المجالين

t = 0

או

0 < t < \frac{π}{2}

0 \leq t < \frac{π}{2}

اتحاد مجالين هو مجموعة الأرقام الموجودة بواحد أو أكثر من المجالين

0 \leq t < \frac{π}{2}

או

\frac{3 π}{2} < t < 2 π

0 \leq t < \frac{π}{2} or \frac{3 π}{2} < t < 2 π

اتحاد مجالين هو مجموعة الأرقام الموجودة بواحد أو أكثر من المجالين

0 \leq t < \frac{π}{2} or \frac{3 π}{2} < t < 2 π

או

t = 2 π

0 \leq t < \frac{π}{2} or \frac{3 π}{2} < t \leq 2 π

0 \leq t < \frac{π}{2} or \frac{3 π}{2} < t \leq 2 π

tan (t) - tan^{2} (t) + sec^{3} (t) :

استخدم دوريّة الـ

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

أمثلة شائعة

-cos(2x)<= (sqrt(3))/2

- cos (2 x) \leq \frac{3}{2}

sin(x)<0,sec(x)>0

sin (x) < 0, sec (x) > 0

2sin(x)+3((sin(2x))/(2sin(x)))<0

2 sin (x) + 3 (\frac{sin ( 2 x )}{2 sin ( x )}) < 0

cos^2(x)>sin(x)cos(x)

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

cos(θ)>0,sin(θ)>0

cos (θ) > 0, sin (θ) > 0