In this video, we will learn how to
identify which quadrant an angle lies and whether its sine, cosine, and tangent will
be positive or negative.
First, let’s consider a coordinate
grid with an 𝑥- and 𝑦-axis. The top-right quadrant is labeled
quadrant one. The top-left quadrant is quadrant
two. The bottom-left quadrant is
quadrant three. And the bottom-right quadrant is
quadrant four. We know to the right of the origin,
the 𝑥-values are positive. And to the left of the origin, the
𝑥-values are negative. In a similar way, above the origin,
the 𝑦-values are positive. And below the origin, the 𝑦-values
are negative. Let’s add four points to our grid:
the point 𝑥, 𝑦; the point negative 𝑥, 𝑦; the point negative 𝑥, negative 𝑦; and
the point 𝑥, negative 𝑦.
And then in the first quadrant, we
draw a line from the origin to the point 𝑥, 𝑦. And we let the angle created
between the 𝑥-axis and this line be 𝜃. If we draw a vertical line from 𝑥,
𝑦 to the 𝑥-axis, we see that we’ve created a right-angled triangle with a
horizontal distance from the origin of 𝑥 and a vertical distance of 𝑦. And if we’re given that it’s one
unit from the origin to the point 𝑥, 𝑦, we can use our trig functions to find out
some things about this triangle.
For our three main trig functions,
sine, cosine, and tangent, the sin of angle 𝜃 will be equal to the opposite side
length over the hypotenuse. The cos of angle 𝜃 will be equal
to the adjacent side length over the hypotenuse. And the tan of angle 𝜃 will be the
opposite side length over the adjacent side length. If we want to find sin of 𝜃, we
can say that it’s equal to 𝑦 over one, since 𝑦 is the opposite side length and the
hypotenuse is one. Similarly, the cosine will be equal
to 𝑥 over one, the adjacent side length over the hypotenuse. And the tan of 𝜃 will be equal to
𝑦 over 𝑥.
We can simplify the sine and cosine
to be 𝑦 and 𝑥, respectively. And because we know that in the
first quadrant all the 𝑦-values are positive, we can say that for angles falling in
quadrant one, the sine value will be positive. Similarly, when we have 𝑥-values
in the first quadrant, we know that the cosine value will also be positive. And tangent in the first quadrant
will be a positive number over a positive number, which will also be positive.
Let’s see how that changes if we
move to the second quadrant. The distance from the origin to
negative 𝑥, 𝑦 is still one. It’s the opposite over the
hypotenuse, 𝑦 over one. But the cosine would then be
negative 𝑥 over one. In the second quadrant, we’re
dealing with negative 𝑥-values, which makes tan of 𝜃 𝑦 over negative 𝑥. This means, in the second quadrant,
the sine relationship remains positive. But the cosine relationship and the
tangent relationship will be negative.
Leaving down to quadrant three,
where we’re dealing with negative 𝑥-coordinates and negative 𝑦-coordinates, sin of
𝜃 will be negative 𝑦 over one. cos 𝜃 is negative 𝑥 over one. We can simplify that to negative 𝑦
and negative 𝑥. But something interesting happens
with tangent. It’s equal to negative 𝑦 over
negative 𝑥, which simplifies to 𝑦 over 𝑥. In the third quadrant, the tangent
relationship is still positive. But in this quadrant, the sine and
cosine relationships will be negative.
And now into the fourth quadrant,
where the 𝑥-coordinate is positive and the 𝑦-coordinate is negative, sin of 𝜃 is
negative 𝑦 over one. But cos of 𝜃 is positive 𝑥 over
one, which gives us a negative sine and a positive cosine. And that will make our tangent
negative 𝑦 over 𝑥. The only positive relationship in
the fourth quadrant is cosine. The negative 𝑦-values make the
sine and tangent relationship negative.
What we discovered for each of
these quadrants will be true for any angle that falls within that quadrant. Any angle in quadrant one will have
positive sine, cosine, and tangent values. Any angle falling in quadrant two
will only have a positive sine relationship. Angles in quadrant three will have
positive tangent relationships. And angles in quadrant four will
have positive cosine relationships.
There is a memory device we
sometimes use to remember this. It’s called the CAST diagram, and
it looks like this. In the CAST diagram, we indicate
which trig relationships are positive in each quadrant. In the fourth quadrant, in the
bottom right, cosine is positive, and sine and tangent are negative. In the first quadrant, in the top
right, we have an A because all three relationships are positive. In the second quadrant, in the top
left, sine is positive, with a negative cosine and a negative tangent. And in the third quadrant, the
bottom left, tangent is positive, and sine and cosine are both negative.
There’s one final thing we need to
review before we look at some examples. And that is how we measure angles
on a coordinate grid. The 𝑥-axis going in the right
direction is called the initial side. And the terminal side is where the
angle stops. If we’re measuring from the initial
side to the terminal side clockwise, we’re measuring a positive angle measure. If we’re measuring from the initial
side to the terminal side in a clockwise manner, we will be measuring a negative
angle measure. And finally, beginning at the
initial side is a measure of zero degrees. From the initial side to the
𝑦-axis is 90 degrees, to the other side of the 𝑥-axis is 180 degrees, 90 degrees
more gets us to 270, and finally back around to 360 degrees.
Now we’re ready to look at some
In which quadrant does the angle
288 degrees lie?
When we think about the four
quadrants of the coordinate grid and label them one through four, we know that the
initial side measures zero degrees. And then each additional quadrant
is 90 more degrees. And then a full rotation is
360. When we measure angles in
coordinate grids, we begin at the 𝑥-axis and proceed in a counterclockwise measure
if we’re dealing with a positive angle. And for us, that means we’ll go
from the initial side, just past 270, since we know that 288 falls between 270 and
360. And we see that this angle is in
the fourth quadrant.
In our next example, we’ll be given
information about the sine and cosine of an angle and asked to find which quadrant
it would lie in.
Determine the quadrant in which 𝜃
lies if cos of 𝜃 is greater than zero and sin of 𝜃 is less than zero.
We’re trying to consider a
coordinate grid and find which quadrant an angle would fall in. We’re told that cos of 𝜃 is
greater than zero, this means it has a positive cosine value, while the sin of 𝜃 is
less than zero, which means the sine has a negative value. One method we use for identifying
the sine and cosine values in different quadrants is the CAST diagram that looks
In the CAST diagram, we know that
in the first quadrant, all values are positive. In the second quadrant, only the
sine value is positive. In the third quadrant, only the
tangent value is positive. And in the fourth quadrant, only
cosine is positive. If we have a negative sine value
and a positive cosine value, we can eliminate quadrant one as all values must be
positive there. We can eliminate quadrant two as
sine is positive there. In quadrant three, sine is
negative, but so is cosine. And that means quadrant three will
not work. In quadrant four, cosine is
positive and sine is negative. And that means our angle 𝜃 under
these conditions must fall in the fourth quadrant.
Let’s consider another example.
In which quadrant does 𝜃 lie if
sin of 𝜃 equals one over the square root of two and cos of 𝜃 equals one over the
square root of two?
When we think about sine and cosine
relationships, we know that sin of 𝜃 is the opposite over the hypotenuse, while the
cos of 𝜃 is the adjacent side over the hypotenuse. But how do we translate that
information into a coordinate grid?
In a coordinate grid, the sine,
cosine, and tangent relationships will have either positive or negative values. And we can remember where each of
these relationships will have positive values with the CAST diagram that looks like
this. In quadrant one, all three trig
relationships are positive. In quadrant two, only the sine
relationship is positive. In quadrant three, only the tangent
relationship is positive. And in quadrant four, only the
cosine relationship is positive.
In this case, we’re dealing with a
positive sine relationship and a positive cosine relationship. We could also use the information
we’re given to find the tangent relationship, which would equal the opposite over
the adjacent. For this angle, that would be one
over one. And what we’re seeing is that all
three of these relationships are positive for this angle. And that means we must say it falls
in the first quadrant.
In our next example, we’ll consider
an angle that’s larger than 360 degrees.
Is cos of 400 degrees positive or
To answer this question, we need to
figure out where 400 degrees would fall on a coordinate grid. If we label our standard coordinate
grid from zero to 360 degrees, we need to think about what we would do with 400
degrees. Traveling counterclockwise one full
rotation, we’ve gone 360 degrees. But in order to get to 400, we’ll
need to go an additional 40 degrees, since 400 minus 360 equals 40. And that means the angle 400 would
fall at the same place that the angle 40 degrees falls, here.
Now we’ve identified where the
angle 400 degrees would be on the coordinate grid, we need to think about how we
would know if this is positive or negative. And to do that, we can use our CAST
diagram that looks like this. Our CAST diagram tells us where
trig relationships are positive in a coordinate grid. In the first quadrant, all three
relationships are positive. In the second quadrant, only sine
is positive. In the third quadrant, only tangent
is positive. And in the fourth quadrant, only
cosine is positive. Our angle falls in the first
quadrant. In the first quadrant, sine,
cosine, and tangent are positive. And that means the cos of 400
degrees will be positive.
Before we finish, let’s review our
key points. We can identify whether sine,
cosine, and tangent will be positive or negative based on the quadrant in which
their angle lies. In quadrant one, the sine, cosine,
and tangent relationships will all be positive. For angles falling in quadrant two,
the sine relationship will be positive, but the cosine and tangent relationships
will be negative. For angles falling in quadrant
three, the sine and cosine relationships will be negative, but the tangent
relationship will be positive. And finally, in quadrant four, the
sine relationship is negative, the cosine relationship is positive, and the tangent
relationship is also negative. We often use the CAST diagram to
remember this. These letters help us identify
which values will be positive in which quadrant.