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>> This is a continuation of Solutions
to the Trig Test 2 review.
Number five is the tangent of 4 positive or negative?
Now, 4 is just a real number.
So when it's a real number, we're thinking
of the circular function.
We have to find, starting over here on the positive side
of the X axis, where we end up, how do we go four units?
Unit circle means the radius is one.
Now, we know if we go to here, that's pi, that's 3.14.
So 4 is just a little bit more than that.
So we're going somewhere about this far,
and that's where it ends.
And so to get the tangent of a function, we have to do the Y
over the X, right, of this ordered pair.
So what's important here is that the X value is negative,
and the Y value is also negative.
So the tangent of 4, I know that it's going to be the Y value,
which is going to be negative,
over the X value, which is negative.
It's going to be a positive number.
So the tangent of 4 is going to be greater than 0.
It's going to be positive for that reason.
So you could write this out, reason, on graph above,
the distance of 4 on the unit circle,
lies in the third quadrant, or quadrant 3,
where the tangent is positive.
Number 6, we're going to graph Y equals negative T sine X.
And we're asked to state the amplitude period phase shift,
if any, and the vertical shift, if any.
So the amplitude is the absolute value
of the coefficient of the function.
So the amplitude of the absolute value of negative 2, which is 2.
So that's the amplitude of the function.
The period.
Well, if it's just an X,
the basic sine function has a period of 2 pi.
If there was a number in front of the X, like 2X,
it would be 2 pi divided by that number, but this is just 2 pi.
And there is no phase shift, because we don't have plus
or minus from that X, and we're not also adding anything
to the entire function.
So there is no phase shift, there is no vertical shift,
so we need to graph this.
So we have to know the basic sine function,
what it looks like, and, whoops, I wanted to put theta instead
of X, just to make sure this is clear, theta,
and what sine theta would be.
And these are the important values to know.
And I remember them by thinking of the unit circle here.
These are the important points here.
So we've got 0 pi over 2 pi, 3 pi over 2, and T pi.
So if we look at these ordered pairs here,
we can compute the sine by looking
at the Y value of each ordered pair.
So the sine of 0 is 0.
The sine of pi over 2 is 1.
The sine of pi is 0 again.
The sine of 3 pi over 2 is negative 1.
And the sine of 2 pi is over here at 0 again.
So you need to know the basic sine function,
and it always breaks up.
We've got 0 to 2 pi, and these are the important points here.
We've got 0 with the first, third and fifth point.
And it goes up one space right here, and down one space there.
And you just make it a smooth curve.
And then remember, this repeats.
So if you went to the other side, if you're going to have
to do two periods, you would just continue this, et cetera.
So you've got the basic sine function.
Now, the difference is we are trying
to do Y equals negative 2.
The sine of X, we're not doing theta,
we're doing X instead of theta.
So I'm just thinking of the X, Y plane here.
And what it means, if there's a minus 2 here, if the amplitude,
it means everything gets stretched.
So it's twice as high.
But the minus sine says it's flipped.
Another way of doing this is simply taking the Y value,
we know what the Y equal sine X is equal to,
if I could just multiply that by 2,
I will know what negative 2 sine X is.
So if I do that, I could take each of these Y values
and multiply it by negative 2.
That's the key.
Multiply by negative 2.
And that will give me, well, 0 times negative 2 is still 0,
1 times negative 2 is negative 2, 0 times negative 2 is 0,
negative 1 times negative 2 is positive 2, and 0.
So if I want, I can write those ordered pairs for my key points,
0 pi over 2 pi, 3 pi over 2, 2 pi, I just have now 0,
negative 2, 0, 2 and 0.
So I'm going to have to go up two and down two on my Y axis.
And my X axis is the same.
All right, my X and Y axis.
And now we're going to be ready to graph this.
So I'm still going to have 0, 0, all these are going to be 0,
but look over here, if I go pi over 2,
I'm going to go down two spaces.
If I want to go to 3 pi over 2, I'm going to go up two spaces.
So the curve, see how it's flipped?
It looks like the one above,
except it goes below the X axis first because of
that minus sine in front of the sine.
And then, of course, I can continue over here.
It just repeats.
So that would be two periods of that function.