Question

Please help!!! Sketch one cycle of the graph of the sine function: y=2sinπθ

Answer 1

You can do a table for several values of tha angle (Ф).

But first you should remember some properties of the sine function.

1) Range: sine function goes from - 1 to + 1.

That means that 2 sin (πФ) goes from - 2 to + 2.

2) sine(0) = 0 => 2sin(π0) = 0

3) sine (π/2) = 1 =>

 Ф = 0.5 => sin(π*0.5) = sin(π/2) = 1

=> 2sin(π*0.5) = 2

3) sin (-π/2) = - 1 =>

Ф = -0.5 => 2sin(-π/2) = -2

4) Periodicity:

The period of sine function is 2π

=> sin (πФ+ 2π) = sin(πФ)

Now you can do the table


Ф       πФ              sin(πФ)                   2sin(πФ)

-1       -π                sin(-π)               2sin(-π) =       0


-5/6    -5π/6           sin(-5π/6)          2sin(-5π/6) =   -1


-1/2     -π/2          sin(-π/2)               2sin(-π/2) =  -2


-1/6    -π/6             sin(-π/6)             2sin(-π/6) =   -1


  0          0               sin(0)              2sin(0) =           0


1/6       π/6              sin(π/6)           2sin(π/6) =       1


1/2       π/2              sin(π/2)           2sin(π/2) =       2                


5/6       5π/6           sin(5π/6)          2sin(5π/6) =     1


1         π                  sin(π)             2sin(π) =           0


Then you can see that one cycle is from Ф = -1/2 to Ф =  1/2


And so you know that you can draw the function in that interval [-1/2, 1/2] and you already have some conspicuos values (from -2 to +2).


You can add some other points to do a better sketch.