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Angles and Trigonometry > Trig Identities

Given the definitions of the taxicab trig functions, the next obvious pursuit would be trig identities. Given the definition of cosine and sine, the first immediate identity comes from the taxicab distance function:

Alternately dividing through by sine and cosine we can obtain the other Pythagorean-style identities:

Using the graphs of cosine and sine we can quickly move on to the basic identities involving even and odd function characteristics and the period of the functions. These identities are very similar to the ones we are familiar with in Euclidean geometry.

 sin(-θ) = -sin(θ) sin(θ + 2) = cos(θ) cos(-θ) = cos(θ) cos(θ - 2) = sin(θ) sin(θ - 4) = -sin(θ) sin(θ + 8k) = sin(θ) cos(θ - 4) = -cos(θ) cos(θ + 8k) = cos(θ)
Table 1: Basic taxicab trigonometric identities

Sum and Difference Formulas

As discussed on the Euclid's Axioms page, SAS congruence for triangles does not hold in modified taxicab geometry. This is simply another way of saying that the Pythagorean Theorem fails. Therefore, the routine proofs of sum and difference formulas are not so routine in this geometry. The first result we will prove is for the cosine of the sum of two angles. The formula given for the cosine of the sum of two angles only takes on two forms; the form used in a given situation depends on the locations of α and β. The notation α ∈ I will be used to indicate α is an angle in quadrant I and similarly for quadrants II, III, and IV.

THEOREM 1: cos(α + β) = ±(-1 + |cosα ± cosβ|) where the signs are chosen to be negative when α and β are on different sides of the x-axis and positive otherwise.

Proof: Without loss of generality, assume α, β ∈ [0, 8), for if an angle θ lies outside [0, 8), ∃k∈Z such that (θ + 8k) ∈ [0, 8) and use of the identity cos(θ + 8k) = cosθ will yield the desired result upon use of the following proof.

All of the subcases have a similar structure. We will prove the subcase α ∈ II, β ∈ III. In this situation 6 ≤ α + β ≤ 10 and we take the negative signs on the right-hand side of the equation. Thus,

The curious case structure in Theorem 1 is due to the odd combinations of quadrants that determine which sign to choose. The reason for the sign change when α and β are on different sides of the x-axis lies in the fact that a corner of the cosine function is being crossed (i.e. different pieces of the cosine function are being used) to obtain the values of the cosine of α and β. Table 2 summarizes which form of cos(α + β) should be used when.

 α β
cos(α + β) = -1 + |cosα + cosβ|
 same quadrant I II III IV
cos(α + β) = 1 - |cosα - cosβ|
 I III I IV II III II IV
Table 2: Forms of cos(α + β) and regions of validity

We can use Theorem 1 and the relations in Table 1 to establish the analogous sine identity.

THEOREM 2: sin(α + β) = ± (-1 + |sinα ± cosβ|) where the signs are chosen to be negative when (α - 2) and β are on different sides of the x-axis and positive otherwise.

Proof: First, note that sinθ = cos(θ - 2). Then,

where in the third line, by Theorem 1, the signs are chosen to be negative when (α - 2) and β are on different sides of the x-axis and positive otherwise.

 α β
sin(α + β) = -1 + |sinα + cosβ|
 I III I IV II II IV IV
sin(α + β) = 1 - |sinα - cosβ|
 I I I II II III II IV III III III IV
Table 3: Forms of sin(α + β) and regions of validity

THEOREM 3: cos(α - β) = ± (-1 + |cosα ± cosβ|) where the signs are chosen to be negative when α and are on different sides of the x-axis and positive otherwise.

Proof:

where in the second line, by Theorem 1, the signs are chosen to be negative when α and are on different sides of the x-axis and positive otherwise.

THEOREM 4: sin(α - β) = ± (-1 + |sinα ± cosβ|) where the signs are chosen to be negative when (α - 2) and are on different sides of the x-axis and positive otherwise.

Proof:

where in the second line, by Theorem 2, the signs are chosen to be negative when (α - 2) and are on different sides of the x-axis and positive otherwise.

Double-angle Formulas

COROLLARY 5: cos(2α) = -1 + 2|cosα|.
(Proof is obvious from Theorem 1.)

COROLLARY 6: sin(2α) = -1 + 2|cos(α - 1)|.

Proof:

References
[1] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.
[2] Akça, Ziya and Rüstem Kaya. On the Taxicab Trigonometry, Journal of Institute Of Mathematics and Computer Sciences, Vol. 10, No. 3, pp. 151-159.

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