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شائع علم المثلثات >

tan(θ)>3cot(θ)

الحلّ

tan (θ) > 3 cot (θ)

الحلّ

\frac{π}{3} + πn < θ < \frac{π}{2} + πn or \frac{2 π}{3} + πn < θ < π + πn

تدوين الفاصل الزمني

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

عشري

1.04719 \dots + πn < θ < 1.57079 \dots + πn or 2.09439 \dots + πn < θ < 3.14159 \dots + πn

خطوات الحلّ

tan (θ) > 3 cot (θ)

انقل

3 cot (θ)

إلى الجانب الأيسر

tan (θ) > 3 cot (θ)

من الطرفين

3 cot (θ)

اطرح

tan (θ) - 3 cot (θ) > 3 cot (θ) - 3 cot (θ)

tan (θ) - 3 cot (θ) > 0

tan (θ) - 3 cot (θ) > 0

tan (θ) - 3 cot (θ)

دوريّة:

π

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

tan (θ), 3 cot (θ)

tan (θ)

دوريّة:

π

π

هي

tan (x)

دوريّة

= π

3 cot (θ)

دوريّة:

π

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

دوريّة ال

π

هي

cot (x)

دوريّة

= \frac{π}{∣ 1 ∣}

بسّط

= π

Combine periods:

π, π

= π

sin, cos :

عبّر بواسطة

tan (θ) - 3 cot (θ) > 0

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

:Use the basic trigonometric identity

\frac{sin ( θ )}{cos ( θ )} - 3 cot (θ) > 0

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

:Use the basic trigonometric identity

\frac{sin ( θ )}{cos ( θ )} - 3 \cdot \frac{cos ( θ )}{sin ( θ )} > 0

\frac{sin ( θ )}{cos ( θ )} - 3 \cdot \frac{cos ( θ )}{sin ( θ )} > 0

\frac{sin ( θ )}{cos ( θ )} - 3 \cdot \frac{cos ( θ )}{sin ( θ )}

بسّط:

\frac{sin ^{2} ( θ ) - 3 cos ^{2} ( θ )}{cos ( θ ) sin ( θ )}

\frac{sin ( θ )}{cos ( θ )} - 3 \cdot \frac{cos ( θ )}{sin ( θ )}

3 \cdot \frac{cos ( θ )}{sin ( θ )}

اضرب بـ:

\frac{3 cos ( θ )}{sin ( θ )}

3 \cdot \frac{cos ( θ )}{sin ( θ )}

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

:اضرب كسور

= \frac{cos ( θ ) \cdot 3}{sin ( θ )}

= \frac{sin ( θ )}{cos ( θ )} - \frac{3 cos ( θ )}{sin ( θ )}

cos (θ), sin (θ)

المضاعف المشترك الأصغر لـ:

cos (θ) sin (θ)

cos (θ), sin (θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (θ)

sin (θ)

= cos (θ) sin (θ)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (θ) sin (θ)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{sin ( θ )}{cos ( θ )} :

multiply the denominator and numerator by

sin (θ)

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

For

\frac{cos ( θ ) \cdot 3}{sin ( θ )} :

multiply the denominator and numerator by

cos (θ)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

\frac{sin ^{2} ( θ ) - 3 cos ^{2} ( θ )}{cos ( θ ) sin ( θ )} > 0

Find the zeroes and undifined points of

\frac{sin ^{2} ( θ ) - 3 cos ^{2} ( θ )}{cos ( θ ) sin ( θ )}

for

0 \leq θ < π

To find the zeroes, set the inequality to zero

\frac{sin ^{2} ( θ ) - 3 cos ^{2} ( θ )}{cos ( θ ) sin ( θ )} = 0

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

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

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

sin^{2} (θ) - 3 cos^{2} (θ) = 0

sin^{2} (θ) - 3 cos^{2} (θ)

حلل إلى عوامل:

(sin (θ) + 3 cos (θ)) (sin (θ) - 3 cos (θ))

sin^{2} (θ) - 3 cos^{2} (θ)

sin^{2} (θ) - (3 cos (θ))^{2}

كـ

sin^{2} (θ) - 3 cos^{2} (θ)

اكتب مجددًا

sin^{2} (θ) - 3 cos^{2} (θ)

a = (a)^{2}

:فعْل قانون الجذور

3 = (3)^{2}

= sin^{2} (θ) - (3)^{2} cos^{2} (θ)

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

(3)^{2} cos^{2} (θ) = (3 cos (θ))^{2}

= sin^{2} (θ) - (3 cos (θ))^{2}

= sin^{2} (θ) - (3 cos (θ))^{2}

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

sin^{2} (θ) - (3 cos (θ))^{2} = (sin (θ) + 3 cos (θ)) (sin (θ) - 3 cos (θ))

= (sin (θ) + 3 cos (θ)) (sin (θ) - 3 cos (θ))

(sin (θ) + 3 cos (θ)) (sin (θ) - 3 cos (θ)) = 0

حلّ كل جزء على حدة

sin (θ) + 3 cos (θ) = 0 or sin (θ) - 3 cos (θ) = 0

sin (θ) + 3 cos (θ) = 0, 0 \leq θ < π : θ = \frac{2 π}{3}

sin (θ) + 3 cos (θ) = 0, 0 \leq θ < π

Rewrite using trig identities

sin (θ) + 3 cos (θ) = 0

cos (θ) \neq = 0, cos (θ)

اقسم الطرفين على

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

بسّط

\frac{sin ( θ )}{cos ( θ )} + 3 = 0

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

:Use the basic trigonometric identity

tan (θ) + 3 = 0

tan (θ) + 3 = 0

انقل

3

إلى الجانب الأيمن

tan (θ) + 3 = 0

من الطرفين

3

اطرح

tan (θ) + 3 - 3 = 0 - 3

بسّط

tan (θ) = - 3

tan (θ) = - 3

tan (θ) = - 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}

θ = \frac{2 π}{3} + πn

θ = \frac{2 π}{3} + πn

0 \leq θ < π :

حلول للمدى

θ = \frac{2 π}{3}

sin (θ) - 3 cos (θ) = 0, 0 \leq θ < π : θ = \frac{π}{3}

sin (θ) - 3 cos (θ) = 0, 0 \leq θ < π

Rewrite using trig identities

sin (θ) - 3 cos (θ) = 0

cos (θ) \neq = 0, cos (θ)

اقسم الطرفين على

\frac{sin ( θ ) - 3 cos ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

بسّط

\frac{sin ( θ )}{cos ( θ )} - 3 = 0

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

:Use the basic trigonometric identity

tan (θ) - 3 = 0

tan (θ) - 3 = 0

انقل

3

إلى الجانب الأيمن

tan (θ) - 3 = 0

للطرفين

3

أضف

tan (θ) - 3 + 3 = 0 + 3

بسّط

tan (θ) = 3

tan (θ) = 3

tan (θ) = 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}

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

0 \leq θ < π :

حلول للمدى

θ = \frac{π}{3}

وحّد الحلول

θ = \frac{2 π}{3}, θ = \frac{π}{3}

Find the undefined points:

θ = \frac{π}{2}, θ = 0

Find the zeros of the denominator

cos (θ) sin (θ) = 0

حلّ كل جزء على حدة

cos (θ) = 0 or sin (θ) = 0

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

cos (θ) = 0, 0 \leq θ < π

cos (θ) = 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}

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

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

0 \leq θ < π :

حلول للمدى

θ = \frac{π}{2}

sin (θ) = 0, 0 \leq θ < π : θ = 0

sin (θ) = 0, 0 \leq θ < π

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn

حلّ:

θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

0 \leq θ < π :

حلول للمدى

θ = 0

وحّد الحلول

θ = \frac{π}{2}, θ = 0

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

ميّز المقاطع المختلفة

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

لخّص في جدول

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

> 0 :

ميّز المقاطع التي تحقّق الشرط

\frac{π}{3} < θ < \frac{π}{2} or \frac{2 π}{3} < θ < π

tan (θ) - 3 cot (θ) :

استخدم دوريّة الـ

\frac{π}{3} + πn < θ < \frac{π}{2} + πn or \frac{2 π}{3} + πn < θ < π + πn

أمثلة شائعة

4tan(x)>4,(-pi/2 , pi/2)

solvefor x,sin(x+30)=tan(10)0<x<360

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cos(y)<0

cos(x)-1>= 2