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شائع علم المثلثات >

tan(x)+cot(x)<1

الحلّ

tan (x) + cot (x) < 1

الحلّ

\frac{π}{2} + πn < x < π + πn

تدوين الفاصل الزمني

(\frac{π}{2} + πn, π + πn)

عشري

1.57079 \dots + πn < x < 3.14159 \dots + πn

خطوات الحلّ

tan (x) + cot (x) < 1

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) < 1

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

:Use the basic trigonometric identity

\frac{sin ( x )}{cos ( x )} + cot (x) < 1

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

:Use the basic trigonometric identity

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

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

\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 )} < 1

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

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}{2}

ميّز المقاطع المختلفة

0 < x < \frac{π}{2}, \frac{π}{2} < 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}{2} + + + + x = \frac{π}{2} + 0 + غير معرّف \frac{π}{2} < x < π + - + - x = π + - 0 غير معرّف

< 0 :

ميّز المقاطع التي تحقّق الشرط

\frac{π}{2} < x < π

tan (x) + cot (x) :

استخدم دوريّة الـ

\frac{π}{2} + πn < x < π + πn

أمثلة شائعة

(tan(x)+1)/(tan(x)-1)<= 0

\frac{tan ( x ) + 1}{tan ( x ) - 1} \leq 0

6sin^2(x)-5sin(x)+1<= 0

6 sin^{2} (x) - 5 sin (x) + 1 \leq 0

2sin(2x)+(1/2)>= 0

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

2cos(2x)-1>= 0

2 cos (2 x) - 1 \geq 0

-cos(x)>=-sin(2x)

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