Of course, we could make a table of values and substitute millions of values of x and join the dots. But this is very dangerous because we don't have a sense of what is going on and our joined dots will look like unrelated spaghetti.

It is best to first sketch the graphs of the 2 parts of this function on the same graph.

`a(x) = 5\ sin\ x` (in blue)

`b(x) = 4\ cos(2x + π/3) ` (in green)

composite trigonometric graph

Now, we need to add the ordinates ( y-values) of each part to obtain the composite graph. Let's take a few values.

When x = 0,

`a(0) = 5\ sin\ 0 = 0`
`b(0) = 4\ cos(π/3) = 2`

Adding these,

a(0) +b(0) = 0 + 2 = 2

Likewise, when x = 1 (radians, of course),

`a(1) = 5\ sin\ 1 = 4. 21`
`b(1) = 4\ cos(2 + π/3) = -3. 98`

Adding gives:

a(1) + b(1) = 0.23

Similarly, for x = 2, a(2) + b(2) = 5.86

x = 3: a(3) + b(3) = 3.59

x = 4: a(4) + b(4) = -7.50

x = 5: a(5) + b(5) = -4.59

Here is the result of adding the 2 separate waves together.

composite trigo graph

 

Notice the two crests near `x = 2.5` have added together to give a large crest (around 7). The two troughs near `x = 4.5` have added together to give a deep trough.