Hyperbolic Functions By Michelle Bunch And Kimberly Duane THE BASICS Let’s Review! “Regular” trig functions are actually called circular functions because each point where x=cosθ and y=sinθ forms the circle x^2+y^2=1 (the unit circle). Hyperbolic functions are similar to circular trig functions. Each point where x=coshα and y=sinhα forms the hyperbola x^2- y^2=1 (the unit hyperbola). Basic Hyperbolic Functions What!? Where did e come from? Let’s find out why! Derivation – Defining stuff • P= point on • α = hyperbolic angle • Derivation is based on defining α based on the area of the sector Derivation - Process 1. Define stuff (what we just did) 2. 3. (Lots of nasty integration, trig substitution, and redefining things in terms of other things) 4. 5. Solve the equation for α in terms of the horizontal and vertical components sinh(x) cosh(x) tanh(x) csch(x) sech(x) coth(x) Hyperbolic Angles • Circular trig functions have an angle θ • Hyperbolic functions’ angles are typically called α • Hyperbolic angles can be written as imaginary circular angles This has a really cool concept associated with it that we’ll cover later! Hyperbolic Functions in the Complex Plane sinh(z) cosh(z) tanh(z) csch(z) sech(z) coth(z) Euler’s Formula Let’s Prove It! • Let’s look at the • Now let’s try the same differential equation y”=-y thing with different • sin(t) and cos(t) are values. solutions • • sin(t)”=(cos(t))’= ______ • • cos(t)”=(-sin(t))’=______ _________________ • All solutions of y”=-y can • take the form: _________________ • _____________________ Euler’s Formula • Therefore, • Solve for A and B. • If t = 0 : • For B, take derivative and then let t = 0. • Euler’s Formula: • Fun Fact: now let t = π – Derive sin(x) and cos(x) • This process uses Euler’s Formula to relate hyperbolic and trig functions. • Let x = it • This same process works for sinh(x). Useful Relations • • • • • • • Note: cosh(x) and sech(x) are even functions like cos(x) and sech(x), while the others are odd. IDENTITIES AND INVERSES Identities (These are all on your handout) Pythagorean Double Angles Sums and Differences Half Angles Particularly Important: Another way to look at it… cosh(x) is the average of sinh(x) is half of the difference of Derive cos(x)cos(y) • Derive arccosh(x). Inverse Functions • • • ; • ; • ; • ; CALCULUS Calculus Derivatives Integrals of Inverse Hyperbolic Derivatives of Inverse Hyperbolic Harder Integrals Taylor Series! Integrals Derivatives! Integrals! Hyperbolic Substitution! • (It works the same way as trig substitution, but you use the basic hyperbolic identities instead of trig identities.) • Try. COOL STUFF Catenaries • A catenary is the shape that a hanging object (like a rope or a chain) makes because of the effect of its mass. • y = cosh(x) • Its volume of revolution is called a catenoid, and it was the first ‘minimal surface’ to be discovered (other than the plane). More Examples Poinsot’s Spirals • These involve graphing hyperbolic functions in polar form. • r = a sech(nθ) • r = a csch(nθ) • They are pretty. .