Trigonometry

Trigonometric identities

Basic identities

  • tanθ=sinθcosθ=ln(cosθ)\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \ln(\cos{\theta})
  • cotθ=cosθsinθ=ln(sinθ)\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \ln(\sin{\theta})

Pythagorean identities

The fundamental Pythagorean Trigonometric identity is:

sinx+cosx=1\sin x + \cos x = 1

From this formula we can derive the formulas for other functions:

Sine and cosine:

sin2x+cos2x=1\sin^2 x + \cos^2 x = 1
  • cos2x=1sin2x\cos^2 x = 1 - \sin^2 x
  • sin2x=1cos2x\sin^2 x = 1 - \cos^2 x

Tangent and secant:

1+tan2x=sec2x1 + \tan^2 x = \sec^2 x
  • tan2x=sec2x1\tan^2 x = \sec^2 x - 1
  • 1=sec2xtan2x1 = \sec^2 x - \tan^2 x

Cotangent and cosecant:

1+cot2x=csc2x1 + \cot^2 x = \csc^2 x
  • cot2x=csc2x1\cot^2 x = \csc^2 x - 1
  • 1=csc2xcot2x1 = \csc^2 x - \cot^2 x