Trigonometry/Trigonometric Identities Reference

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[edit] Pythagoras

  1. sin2(x) + cos2(x) = 1
  2. 1 + tan2(x) = sec2(x)
  3. 1 + cot2(x) = csc2(x)

These are all direct consequences of Pythagoras's theorem.

[edit] Sum/Difference of angles

  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)}

[edit] Product to Sum

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

[edit] Sum and difference to product

  1. Asin(x) + Bcos(x) = Csin(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}

[edit] Double angle

  1. cos(2x) = cos2(x) − sin2(x) = 2cos2(x) − 1 = 1 − 2sin2(x)
  2. sin(2x) = 2sin(x)cos(x)
  3. \tan(2x)=\frac{2\tan(x)}{1- \tan^2(x)}

These are all direct consequences of the sum/difference formulae

[edit] Half angle

  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)}}

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.

[edit] Power Reduction

  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}

[edit] Even/Odd

  1. sin( − θ) = − sin(θ)
  2. cos( − θ) = cos(θ)
  3. tan( − θ) = − tan(θ)
  4. csc( − θ) = − csc(θ)
  5. sec( − θ) = sec(θ)
  6. cot( − θ) = − cot(θ)

[edit] Calculus

  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

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