Trigonometric Functions Summary

Radian Measures

  • 360∘=2π360^\circ = 2\pi radians

Trigonometric Functions

  • Functions: sin, cos, tan, csc, sec, cot
  • Values for angles: π6\frac{\pi}{6}, π4\frac{\pi}{4}, π3\frac{\pi}{3}
  • Pythagoras Theorem: a2+b2=c2a^2 + b^2 = c^2

Unit Circle

  • A circle of radius 1 centered at (0,0).
  • For any angle θ\theta, the y and x coordinates of the point P define the values of sin(θ)sin(\theta), and cos(θ)cos(\theta).

Tangent Function

  • tan(θ)=sin(θ)cos(θ)tan(\theta) = \frac{sin(\theta)}{cos(\theta)}
  • The tangent function(tan) represents the slope of the line from the origin to the point P in the unit circle.

SOH CAH TOA

  • sin(θ)=OppositeHypotenusesin(\theta) = \frac{Opposite}{Hypotenuse}
  • cos(θ)=AdjacentHypotenusecos(\theta) = \frac{Adjacent}{Hypotenuse}
  • tan(θ)=OppositeAdjacenttan(\theta) = \frac{Opposite}{Adjacent}

Inverse Trig Functions

  • sec(θ)=1cos(θ)sec(\theta) = \frac{1}{cos(\theta)} (secant)
  • cosec(θ)=1sin(θ)cosec(\theta) = \frac{1}{sin(\theta)} (cosecant)
  • cot(θ)=1tan(θ)=cos(θ)sin(θ)cot(\theta) = \frac{1}{tan(\theta)} = \frac{cos(\theta)}{sin(\theta)} (cotangent)

Graphs of Trig Functions

  • Sine: y-value
  • Cosine: x-value
  • Tangent: slope

Periodicity

  • Sine: sin(x+2π)=sin(x)sin(x + 2\pi) = sin(x), Range = [-1,1]
  • Cosine: cos(x+2π)=cos(x)cos(x + 2\pi) = cos(x), Range = [-1,1]
  • Tangent: tan(x+π)=tan(x)tan(x + \pi) = tan(x), Range = R

Zeroes/Asymptotes

  • Zeroes (sin) = {nπ∣n∈Zn\pi | n \in Z} = zeroes (tan)
  • Zeroes (cos) = {π2+nπ,n∈Z\frac{\pi}{2} + n\pi, n \in Z} = asymptotes (tan)

Trig Identities

  • cos2(θ)+sin2(θ)=1cos^2(\theta) + sin^2(\theta) = 1
  • 1+tan2(θ)=sec2(θ)1 + tan^2(\theta) = sec^2(\theta)