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شائع >

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

الحلّ

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

الحلّ

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

درجات

x = 0^{\circ} + 6 0^{\circ} n, x = 3 0^{\circ} + 6 0^{\circ} n

خطوات الحلّ

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

sin, cos :

عبّر بواسطة

tan (2 x) + tan (4 x)

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

:Use the basic trigonometric identity

= \frac{sin ( 2 x )}{cos ( 2 x )} + tan (4 x)

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

:Use the basic trigonometric identity

= \frac{sin ( 2 x )}{cos ( 2 x )} + \frac{sin ( 4 x )}{cos ( 4 x )}

\frac{sin ( 2 x )}{cos ( 2 x )} + \frac{sin ( 4 x )}{cos ( 4 x )}

بسّط:

\frac{sin ( 2 x ) cos ( 4 x ) + sin ( 4 x ) cos ( 2 x )}{cos ( 2 x ) cos ( 4 x )}

\frac{sin ( 2 x )}{cos ( 2 x )} + \frac{sin ( 4 x )}{cos ( 4 x )}

cos (2 x), cos (4 x)

المضاعف المشترك الأصغر لـ:

cos (2 x) cos (4 x)

cos (2 x), cos (4 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (2 x)

cos (4 x)

= cos (2 x) cos (4 x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (2 x) cos (4 x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

cos (4 x)

\frac{sin ( 2 x )}{cos ( 2 x )} = \frac{sin ( 2 x ) cos ( 4 x )}{cos ( 2 x ) cos ( 4 x )}

For

\frac{sin ( 4 x )}{cos ( 4 x )} :

multiply the denominator and numerator by

cos (2 x)

\frac{sin ( 4 x )}{cos ( 4 x )} = \frac{sin ( 4 x ) cos ( 2 x )}{cos ( 4 x ) cos ( 2 x )}

= \frac{sin ( 2 x ) cos ( 4 x )}{cos ( 2 x ) cos ( 4 x )} + \frac{sin ( 4 x ) cos ( 2 x )}{cos ( 4 x ) cos ( 2 x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ( 2 x ) cos ( 4 x ) + sin ( 4 x ) cos ( 2 x )}{cos ( 2 x ) cos ( 4 x )}

= \frac{sin ( 2 x ) cos ( 4 x ) + sin ( 4 x ) cos ( 2 x )}{cos ( 2 x ) cos ( 4 x )}

\frac{cos ( 2 x ) sin ( 4 x ) + cos ( 4 x ) sin ( 2 x )}{cos ( 2 x ) cos ( 4 x )} = 0

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

cos (2 x) sin (4 x) + cos (4 x) sin (2 x) = 0

Rewrite using trig identities

cos (2 x) sin (4 x) + cos (4 x) sin (2 x)

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

:فعّل متطابقة الجمع لزوايا

= sin (2 x + 4 x)

sin (2 x + 4 x) = 0

sin (2 x + 4 x) = 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}

2 x + 4 x = 0 + 2 πn, 2 x + 4 x = π + 2 πn

2 x + 4 x = 0 + 2 πn, 2 x + 4 x = π + 2 πn

2 x + 4 x = 0 + 2 πn

حلّ:

x = \frac{πn}{3}

2 x + 4 x = 0 + 2 πn

2 x + 4 x = 6 x :

اجمع العناصر المتشابهة

6 x = 0 + 2 πn

0 + 2 πn = 2 πn

6 x = 2 πn

6

اقسم الطرفين على

6 x = 2 πn

6

اقسم الطرفين على

\frac{6 x}{6} = \frac{2 πn}{6}

بسّط

x = \frac{πn}{3}

x = \frac{πn}{3}

2 x + 4 x = π + 2 πn

حلّ:

x = \frac{π}{6} + \frac{πn}{3}

2 x + 4 x = π + 2 πn

2 x + 4 x = 6 x :

اجمع العناصر المتشابهة

6 x = π + 2 πn

6

اقسم الطرفين على

6 x = π + 2 πn

6

اقسم الطرفين على

\frac{6 x}{6} = \frac{π}{6} + \frac{2 πn}{6}

بسّط

x = \frac{π}{6} + \frac{πn}{3}

x = \frac{π}{6} + \frac{πn}{3}

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

رسم

أمثلة شائعة

2sin(θ)-sqrt(3)=0,0<= θ<2pi

2 sin (θ) - 3 = 0, 0 \leq θ < 2 π

arctan(θ)=-1

arctan (θ) = - 1

tan^2(θ)-tan(θ)-2=0

tan^{2} (θ) - tan (θ) - 2 = 0

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

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

tan(x)=a,tan(x+180)

tan (x) = a, tan (x + 18 0^{\circ})