March 18, 2016
Hiroshi Michiwaki
General Trigonometric Functions and Division by Zero
Even though it would typically be proper to follow the process of definition, theorem, and proof, thedefinitions are explained in detail in the proof below. Consequently, the assertions of the following
theorems make use of the definitions in the proof.
Theorem 1 For the extended general cosine function cosᵣ 𝜃, where 𝜃 is the radial angle, with respect
to the x-axis, of a certain radius vector 𝑟 extending from the origin O on an xy-coordinate plane, thefollowing holds true:

Proof For a certain radius 𝑟 extending from the origin O on an xy-coordinate plane, if 𝜃 is its angle
with respect to the x-axis, its cosine is expressed asIf now 𝑟 ≠ 0 and 𝜃 is defined as 0 ≦ 𝜃𝑟 ≠ 0 = π/2 ∨ 3π/2<2π, then, x𝑟 ≠ 0 ≠ 0. Thus,

Furthermore, if 𝜃 is defined as 0≦ 𝜃𝑟 ≠ 0 = 0 < 2π,However, as 𝑟 = 0 ⇒ x𝑟 = 0 = 0 ∧ 𝜃𝑟 = 0 = 0; if 𝑟 = 0, then the left-hand side of (1) becomes

and the right-hand side becomesThis result depends on the fundamental theorem of division by zero, 0/0 = 0. Therefore, (4) and (5)imply thatThat is, by (3) and (6), we have

Equations (4), (5), and (6) assert thatand (8) gives the impression that the value of the zero cosine function (cos0) could be either 0 or 1. Inpractice, in addition to the value of 𝜃, whether the function takes the value of 0 or 1 depends on theradius 𝑟, which is linearly independent of 𝜃. Thus, it is not considered a single-variable function. Inaddition, as 𝜃 is sometimes dependent on 𝑟, the notation 𝜃ᵣ is considered inappropriate. Therefore, byletting

and considering only the result of (a), i.e., 0 on the rightmost side, it is impossible to determine if it
was obtained when 𝑟 = 0 or 𝑟 ≠ 0. Appending a subscript to the result to leave a trace of 𝑟 and make itpossible to determine how the result was derived then yields

If the notation in (b) is used, there is a certain rationality to the fact that it can be determined whether
𝑟 = 0 or 𝑟 ≠ 0 by mere inspection of the result of the equation, i.e., 00 on the rightmost side. This mayalso give the impression that there are different types of 0.
For the above reasons, although trigonometric functions have so far been treated as one-variable
functions of 𝜃, they are actually two-variable functions of 𝑟 and 𝜃. In the sense discussed here,
trigonometric functions have so far been considered only in cases fulfilling the condition 𝑟 ≠ 0. Thus,
trying to avoid the problem of non-uniqueness by appending a subscript to variables will arguably
result in some degree of inconvenience.
Accordingly, the following is an extension of the definition of the notation of trigonometric functions.
For a radius 𝑟, the sine, cosine, and tangent of the angle 𝜃 between 𝑟 and the reference line areexpressed as

respectively. Thus, when 𝑟 ≠ 0,and 𝑟 = 0 ⇒ 𝜃 = 0, in which case

However, the result of (11) depends on the fundamental theorem of division by zero, 0/0 = 0.
Consequently, (10) and (11) imply thatwhen 𝜃 = 0.
QED.
Theorem 2 For the extended general sine function sinᵣ 𝜃, where 𝜃 is the radial angle, with respect tothe x-axis, of a certain radius vector 𝑟 extending from the origin O on an xy-coordinate plane, thefollowing holds true:

Proof Using (11) and applying 𝑟 = 0 ⇒ 𝜃 = 0 to the definition of the general sine function in (9) yields

Furthermore, using (10), when 𝑟 ≠ 0, and letting 𝜃 = 0 yields

Consequently, based on (13) and (14), the value of the sine function is 0 when 𝜃=0, independent of
the value of 𝑟. Therefore,QED.
Theorem 3 For the extended general tangent function tanᵣ 𝜃, where 𝜃 is the radial angle, with respect
to the x-axis, of a certain radius vector 𝑟 extending from the origin O on an xy-coordinate plane, thefollowing holds true:

Proof Using (11) and applying 𝑟=0 ⇒ 𝜃=0 to the definition of the general tangent function in (9)yieldsand by Theorems 1 and 2, it is clear that

Furthermore, using (10), when 𝑟 ≠ 0, and letting 𝜃=0 yields

Consequently, by (17) and (18), the value of the tangent function is 0 when 𝜃 = 0, independent of thevalue of 𝑟. Therefore,

QED.