como puedo demostra las identidades auxiliares de identidades trigonometricas de angulos compuestos (77puntos)
Respuestas
Respuesta dada por:
1
conociendo las funciones de la suma y diferencia de ángulos, los más importantes son:
![\sin (\alpha \pm\beta )=\sin \alpha \cos \beta \pm \sin \beta \cos \alpha\\ \\
\cos (\alpha \pm\beta )=\cos \alpha \cos \beta \mp \sin \alpha\sin \beta \\ \\
\sin (\alpha \pm\beta )=\sin \alpha \cos \beta \pm \sin \beta \cos \alpha\\ \\
\cos (\alpha \pm\beta )=\cos \alpha \cos \beta \mp \sin \alpha\sin \beta \\ \\](https://tex.z-dn.net/?f=%5Csin+%28%5Calpha+%5Cpm%5Cbeta+%29%3D%5Csin+%5Calpha+%5Ccos+%5Cbeta+%5Cpm+%5Csin+%5Cbeta+%5Ccos+%5Calpha%5C%5C+%5C%5C%0A%5Ccos+%28%5Calpha+%5Cpm%5Cbeta+%29%3D%5Ccos+%5Calpha+%5Ccos+%5Cbeta+%5Cmp+%5Csin+%5Calpha%5Csin+%5Cbeta+%5C%5C+%5C%5C%0A)
![\displaystyle
\tan(\alpha \pm \beta)=\frac{\tan \alpha \pm \tan \beta}{1\mp \tan \alpha \tan \beta} \displaystyle
\tan(\alpha \pm \beta)=\frac{\tan \alpha \pm \tan \beta}{1\mp \tan \alpha \tan \beta}](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Ctan%28%5Calpha+%5Cpm+%5Cbeta%29%3D%5Cfrac%7B%5Ctan+%5Calpha+%5Cpm+%5Ctan+%5Cbeta%7D%7B1%5Cmp+%5Ctan+%5Calpha+%5Ctan+%5Cbeta%7D)
además de la identidad pitagórica
![\sin^2\theta +\cos^2\theta =1 \sin^2\theta +\cos^2\theta =1](https://tex.z-dn.net/?f=%5Csin%5E2%5Ctheta+%2B%5Ccos%5E2%5Ctheta+%3D1)
=========================================================
Así por ejemplo tenemos
(1)
![\displaystyle
\tan 2x=\tan(x+x)=\frac{\tan x+\tan x}{1-\tan x \cdot \tan x}\\ \\
\boxed{\tan 2x=\frac{2\tan x}{1-\tan^2x}} \displaystyle
\tan 2x=\tan(x+x)=\frac{\tan x+\tan x}{1-\tan x \cdot \tan x}\\ \\
\boxed{\tan 2x=\frac{2\tan x}{1-\tan^2x}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Ctan+2x%3D%5Ctan%28x%2Bx%29%3D%5Cfrac%7B%5Ctan+x%2B%5Ctan+x%7D%7B1-%5Ctan+x+%5Ccdot+%5Ctan+x%7D%5C%5C+%5C%5C%0A%5Cboxed%7B%5Ctan+2x%3D%5Cfrac%7B2%5Ctan+x%7D%7B1-%5Ctan%5E2x%7D%7D+)
(2)
![\sin 2x = \sin(x+x)=\sin x\cos x + \sin x \cos x\\ \\
\boxed{\sin 2x = 2\sin x \cos x} \sin 2x = \sin(x+x)=\sin x\cos x + \sin x \cos x\\ \\
\boxed{\sin 2x = 2\sin x \cos x}](https://tex.z-dn.net/?f=%5Csin+2x+%3D+%5Csin%28x%2Bx%29%3D%5Csin+x%5Ccos+x+%2B+%5Csin+x+%5Ccos+x%5C%5C+%5C%5C%0A%5Cboxed%7B%5Csin+2x+%3D+2%5Csin+x+%5Ccos+x%7D)
(3)
![\cos 2x = \cos(x+x)=\cos x\cos x-\sin x \sin x\\ \\
\boxed{\cos 2x =\cos^2x-\sin^2x} \cos 2x = \cos(x+x)=\cos x\cos x-\sin x \sin x\\ \\
\boxed{\cos 2x =\cos^2x-\sin^2x}](https://tex.z-dn.net/?f=%5Ccos+2x+%3D+%5Ccos%28x%2Bx%29%3D%5Ccos+x%5Ccos+x-%5Csin+x+%5Csin+x%5C%5C+%5C%5C%0A%5Cboxed%7B%5Ccos+2x+%3D%5Ccos%5E2x-%5Csin%5E2x%7D)
(4)
![\sin 3x = \sin (2x+x)\\ \\
\sin 3x = \sin 2x \cos x +\sin x \cos 2x\\ \\
\sin 3x = (2\sin x\cos x) \cos x +\sin x (\cos^2 x-\sin^2x)\\ \\
\sin 3x = 2\sin x\cos^2 x +\sin x (\cos^2 x-\sin^2x)\\ \\
\sin 3x = 2\sin x(1-\sin^2x) +\sin x [(1-\sin^2x)-\sin^2x]\\ \\
\sin 3x = 2\sin x-2\sin^3 x+\sin x(1-2\sin^2x)\\ \\
\sin 3x = 2\sin x-2\sin^3 x+\sin x-2\sin^3x\\ \\
\boxed{\sin 3x =3\sin x-4\sin^3x}
\sin 3x = \sin (2x+x)\\ \\
\sin 3x = \sin 2x \cos x +\sin x \cos 2x\\ \\
\sin 3x = (2\sin x\cos x) \cos x +\sin x (\cos^2 x-\sin^2x)\\ \\
\sin 3x = 2\sin x\cos^2 x +\sin x (\cos^2 x-\sin^2x)\\ \\
\sin 3x = 2\sin x(1-\sin^2x) +\sin x [(1-\sin^2x)-\sin^2x]\\ \\
\sin 3x = 2\sin x-2\sin^3 x+\sin x(1-2\sin^2x)\\ \\
\sin 3x = 2\sin x-2\sin^3 x+\sin x-2\sin^3x\\ \\
\boxed{\sin 3x =3\sin x-4\sin^3x}](https://tex.z-dn.net/?f=%5Csin+3x+%3D+%5Csin+%282x%2Bx%29%5C%5C+%5C%5C%0A%5Csin+3x+%3D+%5Csin+2x+%5Ccos+x+%2B%5Csin+x+%5Ccos+2x%5C%5C+%5C%5C%0A%5Csin+3x+%3D+%282%5Csin+x%5Ccos+x%29+%5Ccos+x+%2B%5Csin+x+%28%5Ccos%5E2+x-%5Csin%5E2x%29%5C%5C+%5C%5C%0A%5Csin+3x+%3D+2%5Csin+x%5Ccos%5E2+x+%2B%5Csin+x+%28%5Ccos%5E2+x-%5Csin%5E2x%29%5C%5C+%5C%5C%0A%5Csin+3x+%3D+2%5Csin+x%281-%5Csin%5E2x%29+%2B%5Csin+x+%5B%281-%5Csin%5E2x%29-%5Csin%5E2x%5D%5C%5C+%5C%5C%0A%5Csin+3x+%3D+2%5Csin+x-2%5Csin%5E3+x%2B%5Csin+x%281-2%5Csin%5E2x%29%5C%5C+%5C%5C%0A%5Csin+3x+%3D+2%5Csin+x-2%5Csin%5E3+x%2B%5Csin+x-2%5Csin%5E3x%5C%5C+%5C%5C%0A%5Cboxed%7B%5Csin+3x+%3D3%5Csin+x-4%5Csin%5E3x%7D%0A)
Y así
además de la identidad pitagórica
=========================================================
Así por ejemplo tenemos
(1)
(2)
(3)
(4)
Y así
Preguntas similares
hace 6 años
hace 6 años
hace 9 años
hace 9 años
hace 9 años
hace 9 años
hace 9 años