Rise to find the period
of the given trigonometric functions.
First we have F of X equals two Sine X minus one half Cosine
two thirds X.
Notice here we have a sumer difference
of Sine and Cosine functions.
We know when we have a single Sine function
or Cosine function.
The period is equal to two Pi divided by B,
where B is the Coefficient of X.
But when we have a sumer difference involving
Sine and Cosine functions, the period is going to equal
the least common multiple of the periods
of each individual trig function.
Let's begin by analyzing the graph
to determine the period of F of X.
Let's start at the Y axis or X equals zero
and work our way to the right to see if we can determine
when the graph starts to repeat itself.
If we focus on the maximums, moving to the right,
we have a relative max, relative max, and absolute max.
Then the pattern starts to repeat
relative max, relative max, absolute max and so on.
So analyzing the X axis,
notice how from zero
to six Pi Radians,
we have the graph of one period of the trig function
as well as from six Pi to 12 Pi.
And therefore the period is equal to six Pi Radians.
Let's go back and see
if we can figure this out algebraically.
Let's find the period of each individual trig function.
Let's find the period of Y equals two Sine X
as well as Y equals negative one half Cosine,
two thirds X
for Y equals two Sine X,
the period is equal to two Pi divided by a B where B is one.
And therefore the period is two Pi.
And then for Y equals negative one half Cosine two thirds X
the period is equal to two Pi divided by two thirds.
What is equal to two Pi over one times the reciprocal
of two thirds? Which is three halves.
simplifying, we have the period is equal to three Pi.
So the period of G of X
is equal to the least common multiple of two Pi
and three Pi,
which is equal to six Pi.
Now, if we need to,
we can list out multiples of two Pi and three Pi
to determine the least common multiple.
Multiples of two Pi or two Pi, four Pi,
six Pi,
eight Pi, and so on.
Multiples of three Pi are three Pi, six Pi,
nine Pi, and so on.
And here we can see the least common multiple
of two Pi and three Pi is six Pi.
Let's take a look at a second example.
We have G of X equals negative three Sine four X
plus two Cosine six X.
Let's first look at this graphically.
Again, let's start the y-axis or X equals zero
Let's see if we can determine how long it takes
for the graph to repeat itself.
You're going to be focused on the maximums.
Moving to the right, we have an absolute max,
a relative max, and then the pattern starts to repeat.
Absolute max, relative max, and so on.
Analyzing the X axis from zero to Pi Radians.
We have the graph of one period.
And then again, the graph starts repeat.
From Pi to two Pi, we have the graph of one period.
And from two Pi to three Pi,
we have the graph of one period.
And now let's go back and figure this out algebraically.
To do this, we find the period
of each individual trig function
where the first function is Y equals negative three,
Sine for X
and Y equals
two Cosine six X.
Once we find these periods,
the period of G of X will be the least common multiple
of the two periods.
So for Y equals negative three, Sine four X,
the period is equal to two Pi divided by B,
where B is four.
Two Pi divided by four,
simplifies the Pi over two or one half Pi.
And the period of Y equals two Cosine six X
is equal to two Pi divided by six,
which simplifies two Pi over three or one third Pi.
So the period of G of X is equal
to the least common multiple of one half Pi
and one third Pi.
Again, if we need to, we can list out multiples.
Multiples of one half Pi,
would be one half Pi,
two halves Pi or Pi,
three halves Pi, four halves Pi,
which would be two Pi and so on.
Multiples of one third Pi would be one third Pi,
two thirds Pi, three thirds Pi or Pi,
four thirds Pi, and so on.
We can see the least common multiple is Pi Radians,
which again is the period of G of X.
I hope you found this helpful.