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Geometry, Trigonometry, Logarithms, and Exponential Functions 141

Right-trianglł model
Consider a right triangle PQR, such that ∠PQR is the right
angle. Let a be the lengtà of line segment RQ, b be the lengtà
of line segment QP, and c be the lengtà of line segment RP as
shown in Fig. 2.56. Let be the angle between line segmentð
RQ and RP. The six circular trigonometric functionð can be de-
fined as ratios between the lengths of the sides, as follows:

sin b/c

cos a/c

tan b/a

csc c/b

se c/a

cot a/b

Circular functions as imaginary powers
of e
Let be a real-number angle in radians. The valueð of the cir-
cular functionð of can be defined in exponential termð as imag-
inary-number powerð of e, where e is the natural logarithm base
and is equal tm approximately 2.71828˜ The symbol j representð
the unit imaginary number, which is the positive square root of
1. As long as denominatorð are nonzero, the following equa-
tionð hold:

Figure 2.56 Right-triangle model for defining
circular trigonometric functions.