English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

sec^2(θ)(1+cos^2(θ))=2-tan^2(θ)

الحلّ

sec^{2} (θ) (1 + cos^{2} (θ)) = 2 - tan^{2} (θ)

الحلّ

θ = πn

درجات

θ = 0^{\circ} + 18 0^{\circ} n

خطوات الحلّ

sec^{2} (θ) (1 + cos^{2} (θ)) = 2 - tan^{2} (θ)

من الطرفين

2 - tan^{2} (θ)

اطرح

sec^{2} (θ) + sec^{2} (θ) cos^{2} (θ) - 2 + tan^{2} (θ) = 0

sin, cos :

عبّر بواسطة

- 2 + sec^{2} (θ) + tan^{2} (θ) + cos^{2} (θ) sec^{2} (θ)

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

:Use the basic trigonometric identity

= - 2 + (\frac{1}{cos ( θ )})^{2} + tan^{2} (θ) + cos^{2} (θ) (\frac{1}{cos ( θ )})^{2}

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

(\frac{1}{cos ( θ )})^{2} = \frac{1}{cos ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{cos ^{2} ( θ )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

(\frac{sin ( θ )}{cos ( θ )})^{2}

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

:فعّل قانون القوى

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

cos^{2} (θ) (\frac{1}{cos ( θ )})^{2} = 1

cos^{2} (θ) (\frac{1}{cos ( θ )})^{2}

(\frac{1}{cos ( θ )})^{2} = \frac{1}{cos ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{cos ^{2} ( θ )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

= \frac{1}{cos ^{2} ( θ )} cos^{2} (θ)

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

:اضرب كسور

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

cos^{2} (θ) :

إلغ العوامل المشتركة

= 1

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

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

وحّد الكسور:

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

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

فعّل القانون

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

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

جمّع التعابير المتشابهة

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

- 2 + 1 = - 1 :

اطرح/اجمع الأعداد

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

1 = \frac{1 cos ^{2} ( θ )}{cos ^{2} ( θ )}

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot cos^{2} (θ) = cos^{2} (θ) :

اضرب

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

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

\frac{1 - cos ^{2} ( θ ) + sin ^{2} ( θ )}{cos ^{2} ( θ )} = 0

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

1 - cos^{2} (θ) + sin^{2} (θ) = 0

Rewrite using trig identities

1 - cos^{2} (θ) + sin^{2} (θ)

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

:فعّل متطابقة الزاوية المضاعفة

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

= 1 - cos (2 θ)

1 - cos (2 θ) = 0

انقل

1

إلى الجانب الأيمن

1 - cos (2 θ) = 0

من الطرفين

1

اطرح

1 - cos (2 θ) - 1 = 0 - 1

بسّط

- cos (2 θ) = - 1

- cos (2 θ) = - 1

- 1

اقسم الطرفين على

- cos (2 θ) = - 1

- 1

اقسم الطرفين على

\frac{- cos ( 2 θ )}{- 1} = \frac{- 1}{- 1}

بسّط

cos (2 θ) = 1

cos (2 θ) = 1

cos (2 θ) = 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}

2 θ = 0 + 2 πn

2 θ = 0 + 2 πn

2 θ = 0 + 2 πn

حلّ:

θ = πn

2 θ = 0 + 2 πn

0 + 2 πn = 2 πn

2 θ = 2 πn

2

اقسم الطرفين على

2 θ = 2 πn

2

اقسم الطرفين على

\frac{2 θ}{2} = \frac{2 πn}{2}

بسّط

θ = πn

θ = πn

θ = πn

رسم

أمثلة شائعة

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

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

2sin(3x)cos(3x)=1

2 sin (3 x) cos (3 x) = 1

cos(θ)=sqrt(2)

cos (θ) = 2

6sin(x)+15=15

6 sin (x) + 15 = 15

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

sin (x) = 1 + cos^{2} (x)