Identidades trigonométricas

Fundamentales (pitagóricas y recíprocas)

  • sin⁡2θ+cos⁡2θ=1\sin^{2}\theta + \cos^{2}\theta = 1
  • 1+tan⁡2θ=sec⁡2θ1 + \tan^{2}\theta = \sec^{2}\theta
  • 1+cot⁡2θ=csc⁡2θ1 + \cot^{2}\theta = \csc^{2}\theta
  • tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}, \quad \cot\theta = \dfrac{\cos\theta}{\sin\theta}
  • sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ\sec\theta = \dfrac{1}{\cos\theta}, \quad \csc\theta = \dfrac{1}{\sin\theta}

Paridad y complementarios

  • sin⁡(−θ)=−sin⁡θ,cos⁡(−θ)=cos⁡θ,tan⁡(−θ)=−tan⁡θ\sin(-\theta) = -\sin\theta, \quad \cos(-\theta) = \cos\theta, \quad \tan(-\theta) = -\tan\theta
  • sin⁡ ⁣(π2−θ)=cos⁡θ,cos⁡ ⁣(π2−θ)=sin⁡θ\sin\!\left(\tfrac{\pi}{2}-\theta\right) = \cos\theta, \quad \cos\!\left(\tfrac{\pi}{2}-\theta\right) = \sin\theta

Suma y diferencia de ángulos

  • sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha\pm\beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta
  • cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta
  • tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan(\alpha\pm\beta) = \dfrac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}

Ángulo doble

  • sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta
  • cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^{2}\theta - \sin^{2}\theta = 2\cos^{2}\theta - 1 = 1 - 2\sin^{2}\theta
  • tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \dfrac{2\tan\theta}{1-\tan^{2}\theta}

Ángulo mitad

  • sin⁡θ2=±1−cos⁡θ2,cos⁡θ2=±1+cos⁡θ2\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1-\cos\theta}{2}}, \quad \cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1+\cos\theta}{2}}
  • tan⁡θ2=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\dfrac{\theta}{2} = \dfrac{1-\cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1+\cos\theta}

Reducción de potencias (para integrar)

  • sin⁡2θ=1−cos⁡2θ2,cos⁡2θ=1+cos⁡2θ2\sin^{2}\theta = \dfrac{1-\cos 2\theta}{2}, \quad \cos^{2}\theta = \dfrac{1+\cos 2\theta}{2}
  • sin⁡3θ=3sin⁡θ−sin⁡3θ4,cos⁡3θ=3cos⁡θ+cos⁡3θ4\sin^{3}\theta = \dfrac{3\sin\theta - \sin 3\theta}{4}, \quad \cos^{3}\theta = \dfrac{3\cos\theta + \cos 3\theta}{4}

Producto a suma

  • sin⁡αcos⁡β=12[sin⁡(α+β)+sin⁡(α−β)]\sin\alpha\cos\beta = \tfrac12\bigl[\sin(\alpha+\beta)+\sin(\alpha-\beta)\bigr]
  • cos⁡αcos⁡β=12[cos⁡(α−β)+cos⁡(α+β)]\cos\alpha\cos\beta = \tfrac12\bigl[\cos(\alpha-\beta)+\cos(\alpha+\beta)\bigr]
  • sin⁡αsin⁡β=12[cos⁡(α−β)−cos⁡(α+β)]\sin\alpha\sin\beta = \tfrac12\bigl[\cos(\alpha-\beta)-\cos(\alpha+\beta)\bigr]

Suma a producto

  • sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin\alpha+\sin\beta = 2\sin\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2}
  • sin⁡α−sin⁡β=2cos⁡α+β2sin⁡α−β2\sin\alpha-\sin\beta = 2\cos\dfrac{\alpha+\beta}{2}\sin\dfrac{\alpha-\beta}{2}
  • cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2\cos\alpha+\cos\beta = 2\cos\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2}
  • cos⁡α−cos⁡β=−2sin⁡α+β2sin⁡α−β2\cos\alpha-\cos\beta = -2\sin\dfrac{\alpha+\beta}{2}\sin\dfrac{\alpha-\beta}{2}

Sustitución de Weierstrass ()

  • sin⁡θ=2t1+t2,cos⁡θ=1−t21+t2\sin\theta = \dfrac{2t}{1+t^{2}}, \quad \cos\theta = \dfrac{1-t^{2}}{1+t^{2}}
  • tan⁡θ=2t1−t2,dθ=21+t2 dt\tan\theta = \dfrac{2t}{1-t^{2}}, \quad d\theta = \dfrac{2}{1+t^{2}}\,dt