Chapter 5

Extending trigonometry to angles of any size

Chapter description
This chapter makes it possible for us to use trig ratios with angles of any size, and looks at the graphs of these trig functions. These are very important in many physical applications, so we look at what happens if we shift them and combine them. We also look at methods of handling trig functions and equations.
The chapter is divided into the following sections.

  1. Giving meaning to trig functions of any size of angle
    1. Extending sin and cos
    2. The graph of y = tan x from 0 to 90 degrees
    3. Defining the sin, cos and tan of angles of any size
    4. How does X move as P moves round its circle?
    5. The graph of tan x for any value of x
    6. Can we find the angle from its sin?
    7. The inverse sin and cos functions -- what are they?
    8. What do the graphs of these two inverse functions look like?
    9. Defining the inverse tan function

  2. The trig reciprocal functions
    1. What are these functions?
    2. Their identities -- tan2x + 1 = sec2x and cot2x + 1 = cosec2x
    3. Some examples of proving other trig identities
    4. What do the graphs of the trig reciprocal functions look like?
    5. Drawing other reciprocal graphs

  3. Building more trig functions from the simplest ones
    1. Stretching, shifting and shrinking trig functions
    2. Relating trig functions to how P moves round its circle and SHM
    3. New shapes from putting together trig functions
    4. Putting together trig functions with different periods

  4. Finding rules for combining trig functions
    1. How else can we write sin (A+B)?
    2. A summary of results for similar combinations
    3. Finding tan (A+B) and tan (A-B)
    4. The rules for sin 2A, cos 2A and tan 2A
    5. How could we find a formula for sin 3A ?
    6. Using sin (A+B) to find another way of writing 4 sin t + 3 cos t
    7. More examples of forms like R sin (t + a)
    8. Going back the other way -- the factor Formulas

  5. Solving trig equations
    1. Laying some useful foundations
    2. Finding solutions for equations in cos x
    3. Finding solutions for equations in tan x
    4. Finding solutions for equations in sin x
    5. Solving equations using R sin (x + a) etc.
You can either look at the contents of Chapter 6
or go back to basic trig for working out vector components
or you can go back to the angle between a line and a plane.
or you can return to my book's home page.