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4 Elementary FunctionsTrigonometric Functions

§4.16 Elementary Properties

See accompanying text
Figure 4.16.1: Quadrants for the angle θ. Magnify
Table 4.16.1: Signs of the trigonometric functions in the four quadrants.
Quadrant sinθ,cscθ cosθ,secθ tanθ,cotθ
I + + +
II +
III +
IV +
Table 4.16.2: Trigonometric functions: quarter periods and change of sign.
x θ 12π±θ π±θ 32π±θ 2π±θ
sinx sinθ cosθ sinθ cosθ ±sinθ
cosx cosθ sinθ cosθ ±sinθ cosθ
tanx tanθ cotθ ±tanθ cotθ ±tanθ
cscx cscθ secθ cscθ secθ ±cscθ
secx secθ cscθ secθ ±cscθ secθ
cotx cotθ tanθ ±cotθ tanθ ±cotθ
Table 4.16.3: Trigonometric functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.
sinθ=a cosθ=a tanθ=a cscθ=a secθ=a cotθ=a
sinθ a (1a2)1/2 a(1+a2)1/2 a1 a1(a21)1/2 (1+a2)1/2
cosθ (1a2)1/2 a (1+a2)1/2 a1(a21)1/2 a1 a(1+a2)1/2
tanθ a(1a2)1/2 a1(1a2)1/2 a (a21)1/2 (a21)1/2 a1
cscθ a1 (1a2)1/2 a1(1+a2)1/2 a a(a21)1/2 (1+a2)1/2
secθ (1a2)1/2 a1 (1+a2)1/2 a(a21)1/2 a a1(1+a2)1/2
cotθ a1(1a2)1/2 a(1a2)1/2 a1 (a21)1/2 (a21)1/2 a