The geometric definition of the trig functions
The definitions of the six basic trigonometry functions
Our starting point here is to look at how we define the
six basec trigonometry (or trig for short) functions. Take a
look at the diagram on the left. We start with a circle and a
pair of perpendicular lines that intersect at the center (point
O) of the circle. From there, we draw a diagonal line
heading up and to the right through point X.
We now have an angle XOC which we will call θ (theta). There are a number of measurements that we can make based on this angle with relation to our circle and how it intersects the lines tangent to the circle that we drew in our diagram. Let's draw some lines in the diagram to show these relationships. The diagram for this appears to the right.
The first line that we'll look at is the line that we get when we draw a straight line down from where the line OX intersects the circle to the line OC. We'll call the point where OX intersects the circle B and the point where our new line intersects OC will be A.
The length of the segment AB (we will write this as
AB so we don't have to keep writing "the length of
the segment") we will call the sine of
θ. We
abbreviate sine as "sin" and we'll call that length sin
θ.
We will also call OA (the green segment) the cosine of θ, or cos θ for short.
The vertical line is a tangent to the circle (a tangent, as you may remember, is a line which touches a curve at only one point). D is the point where our diagonal line meets the tangent line and C is where the tangent line meets the circle (and our horizontal line). We call CD (the purple line segment) the tangent of θ or tan θ for short.
The yellow line segment OD is the secant of θ.
We have a second line in the diagram which is also a tangent to the circle. We'll use the prefix "co-" to distinguish the measures that are associated with this line. The orange segment, EF, we will call the cotangent of θ or cot θ for short. The blue segment OF (if you look closely you'll see the edges of this segment peeking out from behind the yellow secant), we'll call the cosecant of θ or csc θ for short.
One last thing about the lengths of the segments that we've drawn here: There are three more segments that we can identify between the labelled points, which are all radii of the circle. If we declare that the radius of our circle is 1, then that tells us that OF=OB=OC=1.
The Pythagorean Theorem and the trig functions
If you look at the diagram that we've drawn, you'll notice that we have three right triangles that appear: OBA, ODC and EOF. And one thing that you should remember about right triangles is the Pythagorean theorem: The the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides: a2+b2=c2.
So what are the lengths of the sides of some of our right
triangles? Let's look first at OBA. The hypotenuse,
OB is a radius, so it has length of 1. We've defined
the sine and cosine as the lengths of AB and
OA respectively. So we can use our knowledge of the
Pythagorean Theorem to find:
sin2 θ + cos2 θ = 1
(Note that we write sin2 θ rather than (sin θ)2 because we will do a lot of writing about squares of trig functions and this saves us writing extra parenthesis. When we read this out loud, we will say "sine squared of theta", which also makes it clear that we mean the square of the sine of theta, rather than the sine of the square of theta).
We can also do this with the triangle ODC (see the diagram at
the right). I'll let you write the relationship yourself as a
homework assignment. You'll also do that with the last right
triangle from the diagram, EOF.
Things we can learn using similar triangles
If you look at the triangles you'll notice the following interesting facts:
- Triangles ΔOBA and ΔODC share an angle at the point O, our angle θ. They also both have right angles, so by angle-angle, they are similar triangles.
- Remember that we only need two angles to know that two triangles are similar, since the third angle comes automatically by the fact that the angles in a triangle add up to 180 degrees (the "no choice" theorem). So the third angle will be 180-(90+θ)=90-θ degrees. And /FOB also has measure 90-θ degrees (since /FOA is a right angle).
- That tells us that the third right triangle, ΔEOF is also similar to the first two. Pretty cool, huh?
But it gets better than that. Remember that we've determined that the individual lengths of the sides are either 1 or one of the six trig functions. So we can use the proportionality of triangles OBA and ODC to find that
|OB|/|OD| = |AB|/|CD| = |OA|/|OC|
and then substitute in the values that we found earlier to find:
From here, we can see things like, cos θ= 1/sec θ or use some simple algebra to solve, for tan θ in terms of the sine and cosine of θ. You'll have a number of homework problems where you'll get to use these relationships.
What happens if the angle is greater than 90 degrees?
Our diagram can be used with angles from 0 to 90 degrees (What
do you think the values for the functions will be for 0
degrees? What about 90 degrees?), but what happens if we want
to look at an angle with measure greater than 90 degrees.
We can take our two perpendicular lines intersecting at the
center of the circle and use them to divide the circle into
four quadrants (see the diagram at the right). The
first quadrant will be angles from 0 to 90 degrees, as we've
already seen. The second quadrant will be angles from 90 to
180 degrees, the third angles from 180 to 270 degrees and the
fourth will be angles from 270 degrees to 360 degrees. If we
go over 360 degrees will be back in the first quadrant, then
the second etc. Similarly, we can have negative angles where
we'll move clockwise around the circle instead of
counterclockwise. So an angle of -30 degrees would be the same
as an angle of 330 degrees and would be in quadrant IV.
If you look to the left, you'll see the diagram that we drew
for the first quadrant redrawn with our angle θ now in
quadrant II. Note, however that the tangent and cotangent
lines haven't moved, so when we need to find the secant and
tangent, we draw the line down and to the right. Because of
the way the lines are drawn, we end up calling some of the
measures negative. Anything that goes down or to the left
compared to quadrant I, we will call negative. For the
diagonal lines OE and OD, we will look to
see whether the line goes forward with the angle line which
will be positive or backwards from the angle line which will
be negative (compare
the secand and cosecant lines in the diagram for quadrant
II). The following table
shows whether we have positive or negative values for each of
the six functions in quadrants I and II. We'll do quadrants
III and IV as homework problems.
| Function | Quadrant I | Quadrant II | ||
|---|---|---|---|---|
| Direction | Sign | Direction | Sign | |
| sine | Up | + | Up | + |
| cosine | Right | + | Left | - |
| tangent | Up | + | Down | - |
| secant | Forwards | + | Backwards | - |
| cotangent | Right | + | Left | - |
| cosecant | Forwards | + | Forwards | + |