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شائع علم المثلثات >

tan(x)-cot(x)>0

الحلّ

tan (x) - cot (x) > 0

الحلّ

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

تدوين الفاصل الزمني

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

عشري

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

خطوات الحلّ

tan (x) - cot (x) > 0

tan (x) - cot (x)

دوريّة:

π

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

tan (x), cot (x)

tan (x)

دوريّة:

π

π

هي

tan (x)

دوريّة

= π

cot (x)

دوريّة:

π

π

هي

cot (x)

دوريّة

= π

Combine periods:

π, π

= π

sin, cos :

عبّر بواسطة

tan (x) - cot (x) > 0

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

:Use the basic trigonometric identity

\frac{sin ( x )}{cos ( x )} - cot (x) > 0

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

:Use the basic trigonometric identity

\frac{sin ( x )}{cos ( x )} - \frac{cos ( x )}{sin ( x )} > 0

\frac{sin ( x )}{cos ( x )} - \frac{cos ( x )}{sin ( x )} > 0

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

بسّط:

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

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

cos (x), sin (x)

المضاعف المشترك الأصغر لـ:

cos (x) sin (x)

cos (x), sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x)

= cos (x) sin (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (x) sin (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sin (x)

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

For

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

multiply the denominator and numerator by

cos (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

Find the zeroes and undifined points of

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

for

0 \leq x < π

To find the zeroes, set the inequality to zero

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

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

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

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

= - cos (2 x)

- cos (2 x) = 0

- 1

اقسم الطرفين على

- cos (2 x) = 0

- 1

اقسم الطرفين على

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

بسّط

cos (2 x) = 0

cos (2 x) = 0

cos (2 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}

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

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

2 x = \frac{π}{2} + 2 πn

حلّ:

x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

2

اقسم الطرفين على

2 x = \frac{π}{2} + 2 πn

2

اقسم الطرفين على

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

بسّط

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

بسّط:

\frac{π}{4} + πn

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{π}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

2 x = \frac{3 π}{2} + 2 πn

حلّ:

x = \frac{3 π}{4} + πn

2 x = \frac{3 π}{2} + 2 πn

2

اقسم الطرفين على

2 x = \frac{3 π}{2} + 2 πn

2

اقسم الطرفين على

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

بسّط

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

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

بسّط:

\frac{3 π}{4} + πn

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

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

: استخدم ميزات الكسور التالية

= \frac{3 π}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{3 π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn

= \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

0 \leq x < π :

حلول للمدى

x = \frac{π}{4}, x = \frac{3 π}{4}

Find the undefined points:

x = \frac{π}{2}, x = 0

Find the zeros of the denominator

cos (x) sin (x) = 0

حلّ كل جزء على حدة

cos (x) = 0 or sin (x) = 0

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

cos (x) = 0, 0 \leq x < π

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}

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

وحّد الحلول

x = \frac{π}{2}, x = 0

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

> 0 :

ميّز المقاطع التي تحقّق الشرط

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

tan (x) - cot (x) :

استخدم دوريّة الـ

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

أمثلة شائعة

sin^2(x)+cos(x)>= 1

sin^{2} (x) + cos (x) \geq 1

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

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

cos(x)-1>= 0

cos (x) - 1 \geq 0

2sin(x/2)-1>0

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

sin(3x)<(sqrt(2))/2

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