Important Trig Concepts Sheet for CSUP Math 124

Here are the trig concepts it is important for you to have in your head at all times:

  1. Definition — if t is any real number and P(x,y) is the corresponding terminal point on the unit circle, then:
    1. sin t =
    2. cos t =
    3. tan t =
    4. csc t =
    5. sec t =
    6. cot t =
    OR: SOHCAHTOA
  2. Some exact values it pays to know:
  3. Odd/even properties — some of the trig functions are odd and some are even. Therefore:
    1. sin(-t) =
    2. cos(-t) =
    3. tan(-t) =
    4. csc(-t) =
    5. sec(-t) =
    6. cot(-t) =
  4. Signs (as in +/-) of the trig functions:
    1. sin t is positive (+) in quadrants ______ and negative (-) in quadrants _______
    2. cos t is positive (+) in quadrants ______ and negative (-) in quadrants _______
    3. tan t is positive (+) in quadrants ______ and negative (-) in quadrants _______
    4. csc t is positive (+) in quadrants ______ and negative (-) in quadrants _______
    5. sec t is positive (+) in quadrants ______ and negative (-) in quadrants _______
    6. cot t is positive (+) in quadrants ______ and negative (-) in quadrants _______
  5. Graphing
    1. algebraically make the function look like a sin(k(x-b)) or a tan(k(x-b))
    2. read off the properties
      1. amplitude =
      2. if working with sin, cos, csc or sec, period =
      3. if working with tan or cot, period =
      4. phase shift =
      5. the "canonical" (standard) full period is the interval =
    3. graph it, based on the above
  6. Identities:
    1. defining identities:
      1. tan t =
      2. csc t =
      3. sec t =
      4. cot t =
    2. the Pythagorean Identity (and variants):
      1. sin2 t + cos2 t = 1
      2. 1 + cot2 t = csc2 t
      3. tan2 t + 1 = sec2 t
    3. New, individual, ad hoc identities... which you have to prove. Strategies for this proof include:
      1. sometimes it pays to rewrite everything in terms of sin and cos;
      2. try some random algebraic operations, such as
        1. put terms over common denominators
        2. factor (techniques like a2-b2=(a-b)(a+b) are often particularly useful) or multiply out
        3. multiply the numerator and denominator of a fraction by the (algebraic, not complex) conjugate of the denominator
        4. use the quadratic formula
      3. the square of any trigonometric formula (such as sin2θ, etc.) is begging for one of the Pythagorean identities to be used; likewise terms with a double or half angle call out for the corresponding identity -- generally, pieces which come from some well-known identity should almost always then be replaced by that identity's remainder
      4. when in doubt, try random (but correct) trig or algebraic identities, in the hopes that something familiar will appear in the wreckage
  7. Addition formulæ:
    1. basic:
      1. sin(s+t) =
      2. cos(s+t) =
    2. be able to get, from the basic formulæ and the even/odd properties, the formulæ for sin(s-t) and cos(s-t).
    3. be able to get, from the basic formulæ, the double angle formulæ (for sin2t, cos2t)
  8. Inverse trig functions:
    1. (confusing!) notation sin-1, etc.
    2. domains are:
      1. for sin-1, domain is __________
      2. for cos-1, domain is __________
      3. for tan-1, domain is __________
    3. it is always true that sin(sin-1(x)) = x, etc.
    4. sometimes sin(sin-1(x)) is not x, instead it is __________________________
    5. how to compute things like cos(sin-1(x)) or other more complex combinations
  9. Solving equations with trig functions in them:
    1. using various identities (and methods from plain ol' algebra -- factoring, quadratic formula, etc.) put all the trig together and on one side of the equation;
    2. there can be at this point several values on the other side of the equation, e.g., if you have used the quadratic formula;
    3. for each of these values, find all values of the unknown angle variable which gives that value of the trig function -- this amounts to:
      1. applying the appropriate inverse trig function
      2. adding "2πk for all integers k" (if the trig function was sin, cos, csc, or sec) or "πk for all integers k" (if the trig function was tan or cot)
      3. possibly simplifying (e.g., sometimes you can use the addendum "πk for all integers k" even for sin, cos, csc, or sec, if the basic solutions before the addendum come in pairs which differ by exactly π).


    10. Jonathan Poritz (jonathan.poritz@gmail.com)
       θ    0      π/6     π/4     π/3     π/2      π   
    sin θ            
    cos θ            
    tan θ