Math 121                                                                                                                        RWN
Trig functions - as easy as 1, 2, 3

Make a table with values n, n/4 and for n =0, 1, 2, 3, 4

                             n            n/4              
                          ---------------------------------------
                             0             0                      0
                             1            1/4                  1/2
                             2            2/4              /2
                             3            3/4              /2
                             4            4/4                    1

Note that the last column is simply the values of sin for the fundamental angles of
0,  π/6, π/4, π/3, π/2, resp. That's all you need to know!

Using the cofunction relationship that cos(x) = sin(π/2 - x), these are just the values of
cos of the same angles, in the reverse order. Thus

                            θ              sin θ            cos θ
                    ---------------------------------------------
                          0                  0                 1
                        π/6              1 / 2          
                        π/4                
                        π/3                     1 / 2
                        π/2                 1                 0

To get values for angles up to π, use your knowledge of the behavior of sin and cos, viz.

Thus
                            θ              sin θ            cos θ
                    ---------------------------------------------
                          0                  0                 1
                        π/6               1/2          
                        π/4                
                        π/3                     1 / 2
                        π/2                 1                 0
                       2π/3                  -1 / 2
                       3π/4               -
                       5π/6             1 / 2         -
                         π                  0                 -1

Similarly for angles up to 2π, use your knowledge of the graphs

To get the values of tan, divide, i.e. tan θ = sin θ / cos θ.
E.g.  tan( 5π/6 ) =  -.

For other  angles use the fact that the periods of sin and cos are 2π, and that of tan is π.
E.g.  sin(99π/4) = sin(24π + 3π/4) = sin(3π/4) =

For negative arguments, you can use the facts that sin and tan are odd and cos is even.

To get values of csc, sec and cot, just take the reciprocals of sin, cos and tan, resp.

Note that now you know the values of all the trig functions for any angle just in
terms of those of sin between 0 and π/4, which in turn are as easy as 1,2,3.

Self Quiz: Pick a trig function at random and an angle which is a multiple of π/6 or π/4.
Compute the value.

Converted by Mathematica  (October 1, 2003)