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Geometry & Trigonometry

Chapter 2 · Identities & Graphs

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Pythagorean identity
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Ratio identity
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
Double angle
sin⁡2θ=2sin⁡θcos⁡θ,cos⁡2θ=cos⁡2θ−sin⁡2θ\sin 2\theta = 2\sin\theta\cos\theta,\quad \cos 2\theta = \cos^2\theta - \sin^2\theta
General sinusoid
y=asin⁡(b(x+c))+dy = a\sin(b(x+c)) + d
amplitude ∣a∣|a|, period 360°/b360°/b, shift cc, midline dd.
Period of tan
tan⁡x\tan x repeats every 180°180° (others every 360°360°).
Solving sin x = k
Solutions xx and 180°−x180° - x, plus 360°360° multiples.
Solving cos x = k
Solutions xx and 360°−x360° - x (i.e. ±x\pm x).
Solving tan x = k
One solution then add 180°180° repeatedly.
Pythagorean
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Double angle
sin⁡2θ=2sin⁡θcos⁡θ,  cos⁡2θ=1−2sin⁡2θ\sin2\theta = 2\sin\theta\cos\theta,\;\cos2\theta = 1-2\sin^2\theta
y = a sin(b(x−c))+d
amp |a|, period 2π/b, horizontal shift c, vertical shift d.
Common trap
Solving trig equations: give ALL solutions in the stated interval (use the period).