Trigonometric Functions and Identities
Identities
Know sin^2θ+cos^2θ=1, 1+tan^2θ=sec^2θ, 1+cot^2θ=csc^2θ; compound angles sin(A±B), cos(A±B), tan(A±B); double angles sin2θ=2sinθcosθ, cos2θ=cos^2θ-sin^2θ=2cos^2θ-1=1-2sin^2θ. General solutions: if sinθ=k then θ=nπ+(-1)^n α; if cosθ=k then θ=2nπ±α; if tanθ=k then θ=nπ+α.
Worked example
Solve 2sin^2x+sinx-1=0, 0≤x≤2π. Factor: (2sinx-1)(sinx+1)=0. So sinx=1/2 gives x=π/6,5π/6; sinx=-1 gives x=3π/2. Check each in the original interval; all three satisfy it.
Graphs and modelling
Reciprocal graphs y=secx, cscx, cotx have asymptotes where the denominator is zero. Radian measure is assumed: arc length s=rθ, sector area ½r^2θ. Harmonic models A sin(B(t-C))+D require amplitude |A|, period 2π/|B|, phase shift C, vertical shift D.
Traps
Dividing by cosx and losing solutions; squaring and not checking extraneous roots; forgetting the ± in cos equations; calculator in degrees; wrong quadrant for inverse trig; using the wrong period; ignoring the given domain.