Trigonometry/For Enthusiasts/Less-Used Trig Identities

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Triangle Identities[edit]

In addition to the Law of Sines, the Law of Cosines, and the Law of Tangents, there are numerous other identities that apply to the three angles A, B, and C of any triangle (where A+B+C=180° and each of A, B, and C is greater than zero). Some of the most notable ones follow:

  1. \displaystyle \cos^2A+\cos^2B+\cos^2C+2\cos A \cos B \cos C=1
  2. \sin A + \sin B + \sin C = 4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}
  3. \displaystyle \tan A + \tan B + \tan C = \tan A \tan B \tan C
  4. \tan \frac{A}{2}\tan\frac{B}{2} + \tan\frac{B}{2}\tan\frac{C}{2} + \tan\frac{C}{2}\tan\frac{A}{2} = 1
  5. \displaystyle \cot A \cot B + \cot B \cot C + \cot C \cot A = 1
  6. \cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2} = \cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2}
  7. \sin A \sin B \sin C = \frac{1}{(\cot A + \cot B)(\cot B + \cot C)(\cot C + \cot A)}
  8. \frac{\sin A + \sin B - \sin C}{\sin A + \sin B + \sin C} = \tan \frac{A}{2}\tan \frac{B}{2}

Pythagoras[edit]

  1. \displaystyle \sin^2(x)+\cos^2(x)=1
  2. \displaystyle 1+\tan^2(x)=\sec^2(x)
  3. \displaystyle 1+\cot^2(x)=\csc^2(x)

These are all direct consequences of Pythagoras's theorem.

Sum/Difference of angles[edit]

  1. \cos(x\pm y)=\cos(x)\cos(y) \mp \sin(x)\sin(y)
  2. \sin(x\pm y)=\sin(x)\cos(y) \pm \sin(y)\cos(x)
  3. \tan(x\pm y)=\frac{\tan(x) \pm \tan(y)}{1 \mp \tan(x) \tan(y)}

Product to Sum[edit]

  1. \displaystyle 2 \sin(x) \sin(y) = \cos(x-y)-\cos(x+y)
  2. \displaystyle 2 \cos(x) \cos(y) = \cos(x-y)+\cos(x+y)
  3. \displaystyle 2 \sin(x) \cos(y) = \sin(x-y)+\sin(x+y)

Sum and difference to product[edit]

  1. \displaystyle A \sin(x)+B\cos(x)= C \sin(x+y), where C=\sqrt{A^2+B^2} and y=\pm\arctan(B/A)
  2. \sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
  3. \sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
  4. \cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
  5. \cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}

Multiple angle[edit]

  1. \cos(2x)=\cos^2(x)- \sin^2(x)=2\cos^2(x)-1=1-2\sin^2(x)
  2. \sin(2x)=2\sin(x)\cos(x)
  3. \tan(2x)=\frac{2\tan(x)}{1- \tan^2(x)}
  4. \cot(2x)=\frac{1}{2}[\cot(x)-\tan(x)]
  5. \csc(2x)=\frac{1}{2}[\cot(x)+\tan(x)]
  6. \cos(3x)=4\cos^{3}(x)-3\cos(x)
  7. \sin(3x)=-4\sin^{3}(x)+3\sin(x)
  8. \tan(3x)=\frac{3\tan(x)-\tan^{3}(x)}{1-3\tan^{2}(x)}
  9. \cos(4x)=8\cos^{4}(x)-8\cos^{2}(x)+1
  10. \sin(4x)=4\sin(x)\cos^{3}(x)-4\sin^{3}(x)\cos(x)
  11. \sin^{2}(4x)=16[\sin^{2}(x)-5\sin^{4}(x)+8\sin^{6}(x)-4\sin^{8}(x)]
  12. \tan(4x)=\frac{4\tan(x)-4\tan^{3}(x)}{1-6\tan^{2}(x)+\tan^{4}(x)}
  13. \cos(5x)=16\cos^{5}(x)-20\cos^{3}(x)+5\cos(x)
  14. \sin(5x)=16\sin^{5}(x)-20\sin^{3}(x)+5\sin(x)
  15. \tan(5x)=\frac{5\tan(x)-10\tan^{3}(x)+\tan^{5}(x)}{1-10\tan^{2}(x)+5\tan^{4}(x)}
  16. \cos(6x)=32\cos^{6}(x)-48\cos^{4}(x)+18\cos^{2}(x)-1
  17. \cos(7x)=64\cos^{7}(x)-112\cos^{5}(x)+56\cos^{3}(x)-7\cos(x)
  18. \sin(7x)=-64\sin^{7}(x)+112\sin^{5}(x)-56\sin^{3}(x)+7\sin(x)
  19. \cos(8x)=128\cos^{8}(x)-256\cos^{6}(x)+160\cos^{4}(x)-32\cos^{2}(x)+1
  20. \cos(nx)=2\cos(x)\cos[(n-1)x]-\cos[(n-2)x]
  21. \sin(nx)=2\cos(x)\sin[(n-1)x]-\sin[(n-2)x]

These are all direct consequences of the sum/difference formulae

Half angle[edit]

  1. \cos(\frac{x}{2})=\pm\sqrt{\frac{1+\cos(x)}{2}}
  2. \sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{2}}
  3. \tan(\frac{x}{2})=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1+\cos(x)}=\pm\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}
  4. \cos^{2}(\frac{3}{2}x)=2\cos^{3}(x)-\frac{3}{2}\cos(x)+\frac{1}{2}

In cases with \pm, the sign of the result must be determined from the value of \frac{x}{2}. These derive from the \cos(2x) formulae.

Power Reduction[edit]

  1. \sin^2\theta=\frac{1-\cos2\theta}{2}
  2. \cos^2\theta=\frac{1+\cos2\theta}{2}
  3. \tan^2\theta=\frac{1-\cos2\theta}{1+\cos2\theta}

Even/Odd[edit]

  1. \sin(-\theta)=-\sin(\theta)
  2. \cos(-\theta)=\cos(\theta)
  3. \tan(-\theta)=-\tan(\theta)
  4. \csc(-\theta)=-\csc(\theta)
  5. \sec(-\theta)=\sec(\theta)
  6. \cot(-\theta)=-\cot(\theta)

Calculus[edit]

  1. \frac{d}{dx}[\sin x] = \cos x
  2. \frac{d}{dx}[\cos x] = -\sin x
  3. \frac{d}{dx}[\tan x] = \sec^{2} x
  4. \frac{d}{dx}[\sec x] = \sec x \tan x
  5. \frac{d}{dx}[\csc x] = -\csc x \cot x
  6. \frac{d}{dx}[\cot x] = -\csc^{2} x