FlyNote SISONKE SANDILE On the Relationships Between arccosx, cosx, and secx. Abstract This paper investigates the relationships between the inverse cosine function arccosx\arccos xarccosx, the cosine function cosx\cos xcosx, and the secant function secx\sec xsecx. We establish identities connecting these functions, derive key lemmas using geometric and algebraic methods, and provide rigorous proofs. Our goal is to clarify how these functions interrelate within the framework of trigonometry and analyze their algebraic expressions. Introduction The inverse cosine function arccosx\arccos xarccosx is fundamental in trigonometry, providing the angle whose cosine is The secant function secx=1cosx\sec x =secx= cosx^-1 is the reciprocal of cosine. Understanding their relationships is essential for solving trigonometric equations and modeling geometric phenomena. This work aims to formalize the connection between arccosx\arccos xarccosx, cosx\cos xcosx, and secx\sec xsecx, exploring identities like: arccosx=xcosxandx2+y2=1,\arccosx=(cos x) x^2 + y^2 = 1, arccosx= cosx and x ^2+y ^2=1, FlyNote and analyzing their algebraic implications. Preliminaries and De fi nitions Cosine Function: cos θ \cos \thetacos θ for θ ∈ R\theta {R} θ ∈ R. Inverse Cosine Function: arccosx\arccos xarccosx, x ∈ [ − 1,1]x \in [-1, 1]x ∈ [ − 1,1], with the principal value arccosx ∈ [0, π ]\arccos x \in [0, \pi]arccosx ∈ [0, π ]. Secant Function: sec θ =1cos θ \sec \theta = {\cos \theta}sec θ = cos θ 1 , de fi ned where cos θ ≠ 0\cos\theta\neq0cos θ =0. Main Results Lemma 1: Geometric Identity of the Unit Circle Lemma: For any point (x,y)(x, y)(x,y) on the unit circle, x2+y2=1.x^2 + y^2 = 1. x ^2 +y ^2=1. Proof: This is a fundamental property of the unit circle: De fi nition:Unit circle={(x,y) ∈ R2 ∣ x2+y2=1 {De fi nition:}{Unit circle} = (x, y) x^2 + y^2 = 1 De fi nition:Unit circle={(x,y) ∈ R x^2+y ^2=1 By the Pythagorean theorem, for any point on the circle: x2+y2=1.x^2 + y^2 = 1. FlyNote x ^2+y ^2 =1. Lemma 2: Relationship Between arccosx\arccos xarccosx and cosx\cos xcosx Lemma: For x ∈ [ − 1,1]x \in [-1, 1]x ∈ [ − 1,1], cos(arccosx)=x,\cos(\arccos x) = x, cos(arccosx)=x, and arccos(cos θ )= θ , θ ∈ [0, π ].\arccos(\cos \theta) = \theta, \quad \theta \in [0, \pi]. arccos(cos θ )= θ , θ ∈ [0, π ]. Proof: The fi rst is the de fi nition of the inverse cosine function. The second follows from the inverse function property restricted to the principal value. Theorem 1: Expression of arccosx in terms of secx Claim: arccosx=sec − 1(1x),arccos x = \sec^-1 arccosx=sec − 1 ( x1), for x ∈ ( − 1,1)x \in (-1, 1)x ∈ ( − 1,1). Proof: FlyNote Starting from θ =arccosx\theta = \arccos x θ =arccosx, then cos θ =x,\cos \theta = x, cos θ =x, and, by the de fi nition of secant, sec θ =1cos θ =1x.\sec \theta = cos\theta= sec θ = cos θ = x1 . Applying the inverse secant, θ =sec − 1(1x),\theta = \sec^(-1) θ =sec − 1 ( x1), which completes the proof. Lemma 3: Algebraic Identity for xxx and yyy on the Unit Circle Lemma: If x=cos θ x = \cos \thetax=cos θ and y=sin θ y = \sin \thetay=sin θ , then: x2+y2=1.x^2 + y^2 = 1. x ^2 +y ^2 =1. Proof: This is the fundamental Pythagorean identity, proved geometrically or algebraically. Main Identity: Connecting secx and the inverse functions FlyNote Claim:arccosx=xcosx(incorrect in general, but explored as an expression).arccos x =cos x(incorrect in general, but explored as an expression arccosx= cosx (incorrect in general, but explored as an expression). However, note that this expression does not hold in general. Instead, the valid identities are: arccosx=sec − 1(1x),\arccos x = \sec^1 arccosx=sec − 1( x1 ), and cos(arccosx)=x. (arccosx)=x.cos(arccosx)=x. Discussion The key takeaway from these lemmas and theorems is that the inverse cosine function can be expressed via the secant function: arccosx=sec − 1(1x) arccosx=sec − 1( x1), for x ∈ ( − 1,1)x \in (-1, 1)x ∈ ( − 1,1). This identity clari fi es the reciprocal relationship between cosine and secant in the context of inverse functions. FlyNote The geometric identity x^2+y^2=x^2 + y^2 = x +y =1 underpins the fundamental relationships between the functions, linking algebraic and geometric perspectives. Conclusion This work formalized the relationships among arccosx, cosx,secx and their identities. The main result demonstrates that: arccosx=sec − 1(1x),arccos=sec^-1x=arccosx=sec − 1( x),valid for x ∈ ( − 1,1) Understanding these identities enhances our ability to solve trigonometric equations and visualize the relationships between these functions. References Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning. Larson, R., & Hostetler, R. (2013). Precalculus with Limits. Cengage Learning. Khan Academy. (n.d.). Inverse Trigonometric Functions. https://www.khanacademy.org/math/trigonometry