FP 3 Chapter 1 Hyperbolic Functions Dr J

FP 3 Chapter 1 Hyperbolic Functions Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 15 th November 2016

Recap of Conic Sections The axis of the parabola is parallel to the side of the cone. All the ones you’ll see can be obtained by taking ‘slices’ of a cone (known as a conic section). C 2: Circles FP 1: Rectangular Hyperbolas FP 1: Parabolas FP 3: Hyperbolas FP 3 Ellipses

Comparing different conics No need to make notes on this yet. We’ll cover it in Chapter 2. We saw circles and ‘rectangular hyperbolas’ back in FP 1. Parabolas Circles Hyperbolas Picture: Wikipedia ? similar ? ar imil s ? ? ? ?

What’s the point of hyperbolas? g! llin e od G m OM ?

Equations for hyperbolic functions ? Say as “shine” ? Say as “cosh” Say as “tanch” ? ? Say as “setch” ? ? Say as “cosetch” Say as “coth”

Equations for hyperbolic functions ? Broculator Tip: Press the ‘hyp’ button. ? ? ?

Exercise 1 2 a b c d ? ? 4 5 ? ? 6 7 3 a b c d 8 ? ? ? ? ?

Sketching hyperbolic functions ? ?

Sketching hyperbolic functions ?

Sketching hyperbolic functions ? ? ?

Test Your Understanding ? ?

Exercise 1 B 1 ? 2 ?

Exercise 1 B 3 ? ? ? ? 4 ? ?

Exercise 1 B 5 ? ?

Hyperbolic Identities ? ? ?

Hyperbolic Identities We can similar prove that: Notice this is + rather than -.

Osborn’s Rule We can get these identities from the normal sin/cos ones by: ? ?

Examples ? ? ?

Exercise 1 C 1 2 3 4 5 6 7 8 9 10 ? ? ?

Exercise 1 C 11 12 ? ? ?

Inverse Hyperbolic Functions As you might expect, each hyperbolic function has an inverse. Note that lack of ‘c’.

Inverse Hyperbolic Functions x

Inverse Hyperbolic Functions ? ? ?

Test Your Understanding ? Show >

Summary Hyperbolic Domain Sketch ? ? Sketch ? 1 1 -1 1 ? ? Inverse Hyperbolic Domain ? ? 1 -1 ? 1

Exercise 1 D 1 2 3 4 5 6 7 8 9 10

Solving Equations Either use hyperbolic identities or basic definitions of hyperbolic functions. ? ?

Test Your Understanding ? ?

Exercise 1 E 1 2 ? ? 3 4 ? ? ? 5 6 7 8 9 10 ?