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Mathematical Expression Editor
Motivating Questions
How can we define trigonometric functions for angles that do not come
from triangles?
Introduction
In the previous sections, you were introduced to the basic trigonometric functions
sine and cosine, and saw how they relate measures of angles to measurements of
triangles. Given a right triangle
we define
There is a limitation in this, which you may have noticed. We can only build a
triangle with a base angle if is between and . We work now to rectify this
deficiency.
The Unit Circle
First, note that the values of sine and cosine do not depend on the scale of the
triangle. Being very explicit, if we take our triangle and scale it up by a factor of
(multiplying each side length by ) we obtain
and Notice that the ratios of the corresponding side lengths are not changed. The
individual side lengths are changed, but the ratios are preserved.
Because of this we could simply assume that whenever we draw a triangle for
computing sine and cosine, that the hypotenuse will have length (by dividing each
side by the length of the hypotenuse). We can do this because we are simply
scaling the triangle, and as we see above, this makes absolutely no difference
when computing sine and cosine. When the hypotenuse is , we find that a
convenient way to think about sine and cosine is via a circle with radius 1.
We call this the unit circle.
The Unit Circle is the circle of radius , with center at
the origin. It is the graph of the equation .
An angle is in standard position if the vertex of the angle is at the origin and one
side oriented along the positive -axis. The ray along the positive -axis is called the
initial side of the angle, and the other ray is called the terminal side of the angle.
The angle can be thought of as the counter-clockwise rotation necessary to spin the
initial side to the terminal side.
Notice that the angles and from the two images above both rotate the initial side of
the angle to the same terminal side, but the angle wraps around the origin first. Two
angles are coterminal if they have the same terminal side. The angles and are
coterminal.
You can also think about an angle wrapping two, three, or four times before getting
to the terminal side. You can also think about rotating clockwise instead of
counter-clockwise. We consider counter-clockwise the positive direction, and
clockwise the negative direction for angles.
Find two angles that are coterminal with , one positive and one negative.
If we start with , and then take another complete counter-clockwise rotation (), we
end up at . That means is coterminal with .
If we start with , and then take another complete clockwise rotation (), we end up at .
That means is coterminal with .
Note that there are many more (in fact, infinitely many) angles coterminal with ,
which can be found by repeatedly adding or subtracting .
Radians
In everyday life, we typically measure angles in degrees. You will see in Calculus that
using degrees can lead to a lot of complications. There is a better choice, more closely
related to the geometry of the circle. Notice that an angle identifies an arc along the
circumerence of the unit circle? We’ll call the arc constructed this way the subtended
arc.
Notice that as grows (counter-clockwise) from the length of this arc, called , also
grows.
If is a right angle, calculate the length .
If , the arc is a quarter of the circle. The circumference of the circle is . Since will be
a quarter of that, .
The units we will be measuring angles, radians, is actually based on arc lengths like
this.
One radian is the angle which, in standard position, subtends an arc of length
.
That is, an angle measuring radian has . Let us suppose that the radius of the circle,
and therefore the length of the subtended arc, has units. This means that an angle of
radian, in a circle of radius unit, subtends an arclength of unit. We know that the
formula for arclength is given by , so that . That means radian is equal to .
Notice that the “units” cancel out? That means radians are a unitless
unit. When our angle is measured in radians, that angle is really just a
number.
In one complete revolution () we have subtended the entire circle so . Based on this
and the definition of the radian above, one complete revolution measures
radians. That is an angle measuring measures radians. This gives us a
way to convert between degrees and radians! Note that can be reduced to .
To convert from radians to degrees, multiply by the factor .
To convert from degrees to radians, multiply by the factor .
Here we’re thinking about as having units . What units do you think should
have?
(a)
Convert , , and to radians.
(b)
Convert radians, radians, and radians to degrees.
(a)
. That means is equivalent to radians.
. That means is equivalent to radians.
. That means is equivalent to radians.
(b)
. That means radians is equivalent to .
. That means radians is equivalent to .
. That means radians is equivalent to .
Frequently we will describe angles by their quadrants. An angle will be called a first
quadrant angle if its terminal side lies in the first quadrant. Any angle in the interval
will be a first quadrant angle, but there are others. For example, is a first quadrant
angle since it is coterminal with . Similarly we will call an angle a second quadrant
angle if its terminal side lies in the second quadrant. These angles are coterminal to
angles with measures . Third quadrant angles and fourth quadrant angles are defined
similarly.
The radian measure of some standard angles are given in the chart below.
Triangles in the Unit Circle
Let’s draw our right triangle from before with the base angle in standard position,
and scaled to have hypotenuse of length . Remember that since the hypotenuse has
length , we know that
and
When we scale our triangle to have hypotenuse of length , sine and cosine measure
the lengths of the opposite and adjacent sides. The triangle in the figure below has its
side lengths labeled with this in mind.
If we consider the hypotenuse of this triangle as terminal side of , the point where
this terminal side intersects the unit circle has coordinates . This has given us
our method to extend trigonometric functions to all angles, instead of just
triangles.
Suppose is an angle in standard position in the unit circle, and denote by the
coordinates of the point where the terminal side of intersects the unit circle.
The domain of sine and cosine is all real numbers, and the other trigonometric
functions are defined precisely when their denominators are nonzero.
Which of the following expressions are equal to ?
(a)
(b)
(c)
(d)
(e)
(f)
Given the angles and intersection point from the definition above:
(a)
so , provided that . This one is correct.
(b)
so , provided that . This is NOT .
(c)
. This is , not .
(d)
, provided that . This one is also correct.
(e)
, provided that BOTH AND . For example, when this fraction is
undefined but . That means is not always the same as .
(f)
, provided that BOTH AND . For example, when this fraction is
undefined but . That means is not always the same as .
Reference Angles
We’ve seen above how to draw a (scaled version of) a right triangle inside the
unit circle, with it’s base angle in standard position. How about the other
way around? If we have an angle that isn’t necessarily an acute angle (one
whose terminal side lies within the first quadrant), would it be possible to
relate it to a triangle? Consider the second quadrant angle in the following
image.
As before, we can draw a vertical line from the point where the terminal side of
intersects the unit circle to the -axis.
No matter the quadrant lies in, we can always construct a triangle by drawing this
vertical side between the -axis and the intersection point. Notice that this triangle
has an acute angle with vertex at the origin.
Suppose is an angle in standard
position. The reference angle, , is the acute angle between the terminal side and
the -axis.
If the terminal side of is along the -axis (in either direction), we will have . However
if the terminal side of lies along the -axis (in either direction) we will have . A
reference angle is never less than , nor greater than .
Find the reference angle for each of the following angles.
(a)
(b)
(c)
(d)
(a)
The angle is between and , so is in the second quadrant.
Since is further (in the counter-clockwise direction) than , we have
(b)
The angle is between and , so is in the third quadrant.
In this case, is further (in the counter-clockwise direction) than . That means
(c)
The angle is coterminal with . Since we see that is a fourth quadrant
angle. Rather than working with , we work with the coterminal angle
.
In this case, is further (in the counter-clockwise direction) than . That means
(d)
Don’t let this one fool you. There is no degree symbol. This angle is not written
as a fraction and it does not seem to include , but that just means is not one of
our standard angles. This measurement is still in radians, though. We are asked
about radians, not degrees. Since is a little more than , we know , so is a first
quadrant angle.
In this case, is further (in the counter-clockwise direction) than . That means
To think about this in a different way, the angle is coterminal with .
Since , it must be its own reference angle. That would mean is the
reference angle of any angles which are coterminal with it, including
.
If we know the reference angle, , can we determine ? Not exactly. In the graph below
are four different angles each having the same reference angle. What’s different about
these angles? They are in different quadrants.
If we know both the reference angle and the quadrant, can we determine the angle?
Not quite. Two angles can have the same quadrant and reference angle if they are
coterminal angles.
We can’t determine the angle exactly, but we can determine the angle’s terminal side.
Since trigonometric functions are given in terms of the coordinates on the terminal
side of the angle, knowing the reference angle and quadrant is enough for
us.
Let’s examine the effects of the quadrants on the trigonometric function values we
discussed earlier.
Quadrant I: and coordinates are both positive so sine values and cosine
values will be positive for these angles.
Quadrant II: is negative and is positive, so cosine values will be negative
and sine values will be positive for these angles.
Quadrant III: and coordinates are both negative so sine values and cosine
values will be negative for these angles.
Quadrant IV: is positive and is negative, so cosine values will be positive
and sine values will be negative for these angles.
Evaluating Trigonometric Functions at Standard Angles
Recall the definition from above.
Suppose is an angle in standard position in the unit
circle, and denote by the coordinates of the point where the terminal side of
intersects the unit circle.
The domain of sine and cosine is all real numbers, and the other trigonometric
functions are defined precisely when their denominators are nonzero.
We are now in a position to evaluate these trig functions for any standard angle.
Calculate the values of the six trigonometric functions for each of the following
angles.
(a)
(b)
(c)
(d)
(a)
is a second quadrant angle with reference angle radians, which is
equivalent to . We know the trig values of : , . In the Quadrant II, -values
are negative and -values are positive so we are looking for a negative cosine
value and a positive sine value. This gives and . Using the definitions of
the other trig functions:
(b)
is a first quadrant angle with reference angle radians, which is equivalent
to . We know the trig values of : , . In the Quadrant I, both the -values
and -values are positive so all trig function values will be positive. This
gives and . Using the definitions of the other trig functions:
(c)
is a third quadrant angle with reference angle radians, which is equivalent
to . We know the trig values of : , . In the Quadrant III, both the -values
and -values are negative so we will have negative sine and cosine values.
This gives and . Using the definitions of the other trig functions:
(d)
has terminal side along the negative -axis (it is coterminal with the angle ).
The terminal side intersects the unit circle at the point .
The coordinates of the intersection give the values of the trig functions and .
Using the definitions of the other trig functions:
Typically we have and , but for the angle we have .
