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شائع علم المثلثات >

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

الحلّ

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

الحلّ

t \in R لا يتحقّق لكلّ

خطوات الحلّ

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

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

tan (t)

دوريّة:

π

π

هي

tan (x)

دوريّة

= 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, 2 π

= 2 π

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

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

بسّط:

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

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

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

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

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

:فعّل قانون القوى

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

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

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

المضاعف المشترك الأصغر لـ:

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

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

Lowest Common Multiplier (LCM)

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

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

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

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

For

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

multiply the denominator and numerator by

cos^{3} (t)

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

For

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

multiply the denominator and numerator by

cos^{2} (2)

\frac{1}{cos ^{3} ( t )} = \frac{1 \cdot cos ^{2} ( 2 )}{cos ^{3} ( t ) cos ^{2} ( 2 )} = \frac{cos ^{2} ( 2 )}{cos ^{2} ( 2 ) cos ^{3} ( t )}

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

= t \in R لا يوجد حلّ لـ

t \in R لا يتحقّق لكلّ

أمثلة شائعة

(2sin^2(2x)-1/2)<= 1/2

(2 sin^{2} (2 x) - \frac{1}{2}) \leq \frac{1}{2}

cos(x)+sqrt(3)*sin(x)>= sqrt(2)

cos (x) + 3 \cdot sin (x) \geq 2

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

cos (\frac{x}{2}) > \frac{2}{2}

tan(4x+pi)<2

tan (4 x + π) < 2

24sin(θ)>0.06

24 sin (θ) > 0.06