Six Basic Trig Functions: Key Points

Derivatives of the six basic trig functions

  • ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos x
  • ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin x
  • ddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x = \sec^2 x
  • ddxcot⁡x=−csc⁡2x\frac{d}{dx}\cot x = -\csc^2 x
  • ddxsec⁡x=sec⁡xtan⁡x\frac{d}{dx}\sec x = \sec x \tan x
  • ddxcsc⁡x=−csc⁡xcot⁡x\frac{d}{dx}\csc x = -\csc x \cot x

Antiderivatives: overview

  • You are expected to memorize all six antiderivatives.
  • Tangent and cotangent antiderivatives are straightforward; secant and cosecant are more involved.
  • Formulas:
  • ∫sin⁡x dx=−cos⁡x+C\int \sin x\,dx = -\cos x + C
  • ∫cos⁡x dx=sin⁡x+C\int \cos x\,dx = \sin x + C
  • ∫tan⁡x dx=−ln⁡∣cos⁡x∣+C=ln⁡∣sec⁡x∣+C\int \tan x\,dx = -\ln|\cos x| + C = \ln|\sec x| + C
  • ∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x\,dx = \ln|\sin x| + C
  • ∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C\int \sec x\,dx = \ln|\sec x + \tan x| + C
  • ∫csc⁡x dx=−ln⁡∣csc⁡x+cot⁡x∣+C\int \csc x\,dx = -\ln|\csc x + \cot x| + C
  • Alternate form for csc: ∫csc⁡x dx=ln⁡∣csc⁡x−cot⁡x∣+C\int \csc x\,dx = \ln|\csc x - \cot x| + C