Several parents and students have asked me if I could post class notes on this site to help when a student misses class.
These are my notes for class--they are very brief.
These notes will not replace the need to attend class but if a student does miss class, this should help him/her find the information missed.
The text book will have much of the necessary information.
Other students from class should be able to fill in any remaining gaps.
When all else fails, ask me.

I apologise for any typos. Please bring them to my attention so that I can fix them.

Trigonometric Equations

section 1

The difference between an expression and an equation
- 'sin x' is an expression
  - 'sin x = .5' is an equation
- expressions can be simplified
  - equations can be solved
- expressions can be evaluated--if you have a value for any variables used in the expression
  - equations can be solved--to find the values that make the equation true
a sample trigonometric equation
- sin x = .5
  - the solutions (and there are more than one)
  - are found by graphing y = sin x and y = .5
  - these 2 graphs coinside in many locations
    - π/6, 5π/6, 13π/6, 17π/6, 25π/6, etc.
    - there are 2 solutions for every 360^o
- specific solutions v. general solutions
  - it is impossible to list all the solutions possible for some trig equations
    - there are 2 convienant methods to solve this problem
      - we put limits on the domain of the trig function and therefore limit the number of possible solutions
        
        normally, you will see something like 0 < θ < 2π
        then you can be expected to list all the specific solutions for the problem
      - you might be asked for all the solutions
        
        you use expressions as π/6 + 2nπ which allows for n to be any integer and therefore this answer would allow for an infinite number of solutions
        this general form of the answer allows you to justify that you have given EVERY solution to the problem
- use the reference angle to find one solution--translate that angle to find all solutions in the given domain
  - the inverse trig function on the calculator will give only one reference solution
  - you then use your knowledge of
    - the repetative nature of the trig functions
    - the period of the function
    - the graphs of the functions
  - in the example sin x = .5
    - you can graph the functions 'y = sin x' and 'y = .5'
    - make sure that your window is set appropriately
      - set xmin and xmax to the domain that you need for the problem
      - set ymin and ymax appropriately for the problem--
        
        I use -2 and 2 for the sin x = .5 example
        this ymin and ymax will work for most of the sec and csc problems too
      - be sure to check the degree radian thing also--maybe you should do this first
    - you can use 'trace' to get close to the angle measures you are looking for
    - the inverse trig functions will give you better details on the answer(s)
When you are asked to 'solve an equation'--
- if there is more than one solution,
- you must give all of the solutions
number of solutions for a trigonometric equation
- there may be an infinite number of solutions
- there may be a limited number of solutions
- there may be only one solution
- there may be no solutions possible
inclination--the angle α
- the angle (measured CCW) formed by the positive x-axis and the line in question
- slope of any line is Δy / Δx
- m = Δy / Δx = sin α / cos α = tan α
  - this formula assumes that α is not 90^o
  - if α = 90^o the line is vertical
  - vertical lines have 'no slope'

section 2

function families
- the sin function is a specific function
  - it has a specific shape
  - it has a y intercept at a predictable location (always the same place)
  - it has x intercepts at predictable locations (always the same places)
  - it is continuous--no breaks or asymptotes
    - this list of characteristics could go on for quite a while
- sin can also be thought of as a family of functions
  - they all have the same basic shape shape--
    - a wave that goes up and comes down
    - the high points are all at the same height
    - the low points are all at the same height
    - etc.
  - they all have their y intercept at a predictable location
  - they all have x intercepts at predictable locations
  - it is continuous--no breaks or asymptotes
- y = sin x
  - is the parent function
  - this is the function in its purest, simplest form
  - this is always a good place to start
- there are many variations of this type of function--try graphing these functions
  - y = 2 sin x
  - y = 3 sin x
  - y = sin 2x
  - y = sin 3x
  - y = - sin x
  - y = (sin x) + 4
  - y = (sin x) - 4
  - y = -1/2 sin 3x
- these equations (and millions of other variations all have the same charactictics of all the other sin functin
- this is called the SIN FAMILY
- general form of the sin family of functions: y = A sin B(x - h) + k
  - A ≠ 0, B > 0
|A| gives you the amplitude
- amplitude indicates how much the function is stetched (or squashed) vertically compared to the parent funtion
- the parent function is always the basis of comparison
  - if A is larger than 1, the function is taller than the standard
  - if A is smaller than 1, the function is shorter than the standard
  - if A is negative, the function is reflected (mirror image) across the x axis
  - the amplitude gets the absolute value sign since amplitude refers to the vertical 'width' of the graph--it does not matter if it 'goes up' first or later
B gives you the period
- period refers to how far along the axis you have to travel before the function begins to repeat
- period is the length of the domain required to complete one cycle of the function
- period is related to frequency (from your science class)
  - period = (2π)/B
    - a B larger than 1 will squeese the graph tighter
      - like compressing a spring
    - a B smaller than 1 will pull the graph out wider
      - like stretching a spring
    - B cannot be negative or zero
any basic function can be the parent of a family
- y = 3 tan .2x
  - belongs to the tangent family
  - you should expect it to have common charactistics with the function y = tan x
- y = 5 - 2sec 3x
  - belongs to the secant family
  - has charactistics common with the function y = sec x
h and k from the general form above will be addressed in the next section

section 3

the parent function of each of the 6 trig functions is balanced nicely above and below the x axis
- some of the family members are not so nicely balance above and below the x-axis
- they still have a horizontal axis--it just may not be the x axis
  - these graphs have been slid up or down from their standard position
the origin coincides with the starting point on the unit circle with the angle measuring 0 being at the point (1,0)
- some members of the sin family do not cross the y axis at the origin
- these graphs have been slid left or right from their standard position
y - k = sin (x - h)
- h controls the horizontal shift
- k controls the vertical shift
- on a calculator you will use the form y = sin (x - h) + k
  - please note the + and the - sign in the formula above
  - the shift is what you would expect from your work with parabolas
    - positive h shifts the graph right
    - negative h shifts the graph left
    - positive k shifts the graph up
    - negative k shifts the graph down
each of the trig functions can be treated in the same way
- they can be shifted up/down by adding in a 'k'
- they can be shifted left/right by inserting a 'h'
y - k = A sin B(x - h)
- with practice, you should be able to look at a graph and tell me the equation that created it
- with practice, you should be able to take any equation and without a calculator, create a decent sketch of the function

section 4

relationships between functions
- some of these you should already know
  - csc θ = 1 / sin θ
  - sec θ = 1 / cos θ
  - cot θ = 1 / tan θ = cos θ / sin θ
- some of them are easy to see--if you think about it for a little while
  - if sin θ = y, then sin -θ = -y since they are mirror images across the x axis on the unit circle
    - this means that sin -θ = -y = - sin θ
    - or just sin -θ = - sin θ
    - if you measure the angle clockwise instead of the normal counter-clockwise, you get the opposite (pos/neg) but the y value remains the same
    - csc -θ behaves the same as sin -θ and therefore
    - csc -θ = -csc θ
  - if cos θ = x, then cos -θ = x since the unit circle shows that both angles have the same x coordinate
    - this means that cos -θ = x = cos θ
    - or just cos -θ = cos θ
    - it does not matter if you choose to measure the angle clockwise or counter clockwise, the cos of that angle is still the same
    - sec -θ behaves the same as cos -θ and therefore
    - sec -θ = sec θ
  - if tan θ = y / x, then tan -θ = -y / x
    - this means that tan -θ = - tan θ
    - if you measure an angle counter clockwise, and you have another angle that was measured clockwise but the same measure, their tangents are the same number but one is positive and the other is negative
    - cot -θ behaves the same as tan -θ and therefore
    - cot -θ = -cot θ
- some relationships between the trig functions involve more than one trig function
  - think about any point on the unit circle
    - from that point, draw a segment straight down
    - draw in the radius
    - you should have a right triangle
  - working with this right triangle
    - the vertical leg measures y units (a.k.a. sin θ)
    - the horizontal leg measures x units (a.k.a. cos θ)
    - the radius = 1 unit
    - the Pythagorean Theorem states leg² + leg = hyp²
    - when you bring this all together you get
      - sin ² θ + cos ² θ = 1
    - if you start with sin ² θ + cos ² θ = 1
    - and you divide each term on both sides of the equation by the same thing you can get 2 more of these Pythagorean Relationships between the trig functions
    - divide by cos ² θ
    - you get:
      - (sin ² θ) / (cos ² θ)+ (cos ² θ) / (cos ² θ) = 1 / (cos ² θ)
      - which simplifiest to:
      - tan ² θ + 1 = sec ² θ
    - start with sin ² θ + cos ² θ = 1 and divide by sin ² θ
    - you get:
      - (sin ² θ) / (sin ² θ)+ (cos ² θ) / (sin ² θ) = 1 / (sin ² θ)
      - which simplifiest to:
      - 1 + cot ² θ = csc ² θ
- please note that these Pythagorean Relationships involve 2 different trig functions in the same equation
  - you do not understand why yet, but this is significant
- these last few relationships involve the fact that the trig functions really act as pairs
  - sine and cosine are the same function (if you leave off the axes, it is hard to tell which is which)
    - you can think of these graphs as basically the same thing but shifted horizontally 90^o from each other
    - in degrees--
      - sin θ = cos (90^o - θ)
      - cos θ = sin (90^o - θ)
      - tan θ = cot (90^o - θ)
      - sec θ = csc (90^o - θ)
      - csc θ = sec (90^o - θ)
      - cot θ = tan (90^o - θ)
        
        try these out on your calculator
        graph the first half of the above equation
        then graph the second half on the same graph
        if you make the second graph a thicker line, you will see clearly if the 2 graphs coincide perfectly
    - in radians--
      - sin θ = cos (&pi/2 - θ)
      - cos θ = sin (&pi/2 - θ)
      - tan θ = cot (&pi/2 - θ)
      - sec θ = csc (&pi/2 - θ)
      - csc θ = sec (&pi/2 - θ)
      - cot θ = tan (&pi/2 - θ)
  - secant and cosecant behave the same way with regard to each other
  - tangent and cotangent are the same shape graphs but mirror images
identities
- identities are basic relationships between functions that are always true over any domain where the 2 functions are defined
- there have been many algebraic identities in your past
- the relationships described above can all be considered trigonometric identities
- proving identities takes a little practice and like in algebra is simply put just a list of algebraic steps

section 5

more interesting trig equations
- some equations (for one reason or another) are more challenging to solve
- you can use trigonometric identities to change from the problem that you are given into an easier problem

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