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شائع علم المثلثات >

sin^2(a)sec(a)=tan^2(a)

الحلّ

sin^{2} (a) sec (a) = tan^{2} (a)

الحلّ

a = 2 πn, a = π + 2 πn

درجات

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

sin^{2} (a) sec (a) = tan^{2} (a)

من الطرفين

tan^{2} (a)

اطرح

sin^{2} (a) sec (a) - tan^{2} (a) = 0

sin, cos :

عبّر بواسطة

- tan^{2} (a) + sec (a) sin^{2} (a)

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

:Use the basic trigonometric identity

= - (\frac{sin ( a )}{cos ( a )})^{2} + sec (a) sin^{2} (a)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:فعّل قانون القوى

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

\frac{1}{cos ( a )} sin^{2} (a) = \frac{sin ^{2} ( a )}{cos ( a )}

\frac{1}{cos ( a )} sin^{2} (a)

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

:اضرب كسور

= \frac{1 \cdot sin ^{2} ( a )}{cos ( a )}

1 \cdot sin^{2} (a) = sin^{2} (a) :

اضرب

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

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

cos^{2} (a), cos (a)

المضاعف المشترك الأصغر لـ:

cos^{2} (a)

cos^{2} (a), cos (a)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos^{2} (a)

cos (a)

= cos^{2} (a)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos^{2} (a)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

cos (a)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

- sin^{2} (a) + cos (a) sin^{2} (a) = 0

- sin^{2} (a) + cos (a) sin^{2} (a)

حلل إلى عوامل:

sin^{2} (a) (cos (a) - 1)

- sin^{2} (a) + cos (a) sin^{2} (a)

sin^{2} (a)

قم باخراج العامل المشترك

= sin^{2} (a) (- 1 + cos (a))

sin^{2} (a) (cos (a) - 1) = 0

حلّ كل جزء على حدة

sin^{2} (a) = 0 or cos (a) - 1 = 0

sin^{2} (a) = 0 : a = 2 πn, a = π + 2 πn

sin^{2} (a) = 0

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

فعّل القانون

sin (a) = 0

sin (a) = 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}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn

حلّ:

a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

انقل

1

إلى الجانب الأيمن

cos (a) - 1 = 0

للطرفين

1

أضف

cos (a) - 1 + 1 = 0 + 1

بسّط

cos (a) = 1

cos (a) = 1

cos (a) = 1 :

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}

a = 0 + 2 πn

a = 0 + 2 πn

a = 0 + 2 πn

حلّ:

a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

وحّد الحلول

a = 2 πn, a = π + 2 πn

رسم

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