Keep exploring at ► https://brilliant.org/TreforBazett. Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription! Today we unpack everything to do with hyperbolic functions. In calculus we often see an analytic definition of hyperbolic cosine and hyperbolic sin expressed as exponential functions. But why? Analytically, this is just asking for the even and odd components of e^x. However, we can connect this geometrically to hyperbolas, and specifically to the area enclosed by a hyperbola. 0:00 Even and Odd Functions 2:24 Analytic Definition of cosh and sinh 3:19 Graphic cosh and sinh 3:53 Taylor series and derivatives 5:12 Hyperbolas 6:42 Defining trig functions geometrically 9:22 Defining hyperbolic trig functions geometrically 10:45 The geometric and analytic definitions are the same 14:03 Euler's Equation 15:06 Brilliant.org/TreforBazett Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://beautifulequations.net/pages/trefor COURSE PLAYLISTS: ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Log...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   ►CALCULUS I:    • Calculus I (Limits, Derivative, Integ...   ► CALCULUS II:    • Calculus II (Integration Methods, Ser...   ►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus ...   ►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line In...   ►DIFFERENTIAL EQUATIONS:    • Ordinary Differential Equations (ODEs)   ►LAPLACE TRANSFORM:    • Laplace Transforms and Solving ODEs   ►GAME THEORY:    • Game Theory   OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0 ►Cool Math Series:    • Cool Math Series   BECOME A MEMBER: ►Join: null MATH BOOKS I LOVE (affilliate link): ► https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography