Digital Library of Mathematical Functions
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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 (1-a2)1/2 a(1+a2)-1/2 a-1 a-1(a2-1)1/2 (1+a2)-1/2
cosθ (1-a2)1/2 a (1+a2)-1/2 a-1(a2-1)1/2 a-1 a(1+a2)-1/2
tanθ a(1-a2)-1/2 a-1(1-a2)1/2 a (a2-1)-1/2 (a2-1)1/2 a-1
cscθ a-1 (1-a2)-1/2 a-1(1+a2)1/2 a a(a2-1)-1/2 (1+a2)1/2
secθ (1-a2)-1/2 a-1 (1+a2)1/2 a(a2-1)-1/2 a a-1(1+a2)1/2
cotθ a-1(1-a2)1/2 a(1-a2)-1/2 a-1 (a2-1)1/2 (a2-1)-1/2 a