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شائع علم المثلثات >

sec(2x)+tan(2x)=2

الحلّ

sec (2 x) + tan (2 x) = 2

الحلّ

x = 0.32175 \dots + πn

درجات

x = 18.43494 \dots^{\circ} + 18 0^{\circ} n

خطوات الحلّ

sec (2 x) + tan (2 x) = 2

من الطرفين

2

اطرح

sec (2 x) + tan (2 x) - 2 = 0

sin, cos :

عبّر بواسطة

- 2 + sec (2 x) + tan (2 x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

\frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )}

وحّد الكسور:

\frac{1 + sin ( 2 x )}{cos ( 2 x )}

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

فعّل القانون

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

1 + sin (2 x) - 2 cos (2 x) = 0

Rewrite using trig identities

1 + sin (2 x) - 2 cos (2 x)

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

:فعّل متطابقة الزاوية المضاعفة

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

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

1 + sin (2 x)

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

:فعّل نطريّة فيتاغوروس

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

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

:فعّل متطابقة الزاوية المضاعفة

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

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

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

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

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

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

بسّط:

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

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

(sin (x) + cos (x))^{2} : sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = sin (x), b = cos (x)

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

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

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

وسٌع:

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

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 2, b = cos^{2} (x), c = sin^{2} (x)

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

فعّل قوانين سالب-موجب

- (- a) = a

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

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

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

بسّط:

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

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

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

اجمع العناصر المتشابهة

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

sin^{2} (x) + 2 sin^{2} (x) = 3 sin^{2} (x) :

اجمع العناصر المتشابهة

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

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

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

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x) = 0

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x)

حلل إلى عوامل:

(3 sin (x) - cos (x)) (sin (x) + cos (x))

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x)

قسّم التعابير لمجموعات

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

تعريف

Factors of

3 : 1, 3

3

Divisors (Factors)

Find the Prime factors of

3 : 3

3

هو عدد أوّليّ لذلك لا يمكن تحليله لعوامل أوّليّة

3

= 3

Add 1

1

3

قواسم

1, 3

Negative factors of

3 : - 1, - 3

Multiply the factors by

- 1

to get the negative factors

- 1, - 3

For every two factors such that

u * v = - 3,

check if

u + v = 2

Check

u = 1, v = - 3 : u * v = - 3, u + v = - 2 \Rightarrow

خطأCheck

u = 3, v = - 1 : u * v = - 3, u + v = 2 \Rightarrow

صحيح

u = 3, v = - 1

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

(3 sin^{2} (x) - sin (x) cos (x)) + (3 sin (x) cos (x) - cos^{2} (x))

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

sin (x) (3 sin (x) - cos (x))

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

من

sin (x)

اخرج العامل

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

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

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

= 3 sin (x) sin (x) - sin (x) cos (x)

sin (x)

قم باخراج العامل المشترك

= sin (x) (3 sin (x) - cos (x))

cos (x) (3 sin (x) - cos (x))

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

من

cos (x)

اخرج العامل

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

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

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

= 3 sin (x) cos (x) - cos (x) cos (x)

cos (x)

قم باخراج العامل المشترك

= cos (x) (3 sin (x) - cos (x))

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

3 sin (x) - cos (x)

قم باخراج العامل المشترك

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

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

حلّ كل جزء على حدة

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

3 sin (x) - cos (x) = 0 : x = arctan (\frac{1}{3}) + πn

3 sin (x) - cos (x) = 0

Rewrite using trig identities

3 sin (x) - cos (x) = 0

cos (x) \neq = 0, cos (x)

اقسم الطرفين على

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

بسّط

\frac{3 sin ( x )}{cos ( x )} - 1 = 0

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

:Use the basic trigonometric identity

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

3 tan (x) - 1 = 0

للطرفين

1

أضف

3 tan (x) - 1 + 1 = 0 + 1

بسّط

3 tan (x) = 1

3 tan (x) = 1

3

اقسم الطرفين على

3 tan (x) = 1

3

اقسم الطرفين على

\frac{3 tan ( x )}{3} = \frac{1}{3}

بسّط

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3}

Apply trig inverse properties

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3} :

حلول عامّة لـ

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

sin (x) + cos (x) = 0 : x = \frac{3 π}{4} + πn

sin (x) + cos (x) = 0

Rewrite using trig identities

sin (x) + cos (x) = 0

cos (x) \neq = 0, cos (x)

اقسم الطرفين على

\frac{sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

\frac{sin ( x )}{cos ( x )} + 1 = 0

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

:Use the basic trigonometric identity

tan (x) + 1 = 0

tan (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

tan (x) + 1 = 0

من الطرفين

1

اطرح

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

بسّط

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1 :

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}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

وحّد الحلول

x = arctan (\frac{1}{3}) + πn, x = \frac{3 π}{4} + πn

\frac{3 π}{4} + πn :

بما أنّ المعادلة غير معرّفة لـ

x = arctan (\frac{1}{3}) + πn

أظهر الحلّ بالتمثيل العشريّ

x = 0.32175 \dots + πn

رسم

أمثلة شائعة

sec(2x)+tan(2x)=9

sec (2 x) + tan (2 x) = 9

tan(4x)-tan(2x)=0

tan (4 x) - tan (2 x) = 0

solvefor t,sin(t)+2sin(2t)=0

so l v e f or t, sin (t) + 2 sin (2 t) = 0

tan^2(x)-sin(x)tan^2(x)=0

tan^{2} (x) - sin (x) tan^{2} (x) = 0

4sin(x)=2.54648

4 sin (x) = 2.54648