Download Trigonometry

Trigonometry
Trigonometry is the study of relationships between the side lengths and angle measures of
triangles (literally, triangle measure). An understanding of trigonometry is useful for further
study in mathematics (e.g., calculus) and vital for applications in many different areas of
science. Knowledge of the relationships between the components of triangles has obvious
application in geography, navigation, optics, astronomy, and basic kinematics, but it is also
exceptionally useful in other non-obvious applications such as acoustics, signal processing,
and seismology, as well as the study of cyclic or periodic phenomena.
Let’s begin our discussion of trigonometry by reviewing some basic facts about triangles.
1.
Triangle Basics
A triangle is a polygon with three sides. Typically, we denote the vertices by upper-case
letters (such as A, B, and C) and the sides by lower-case letters (such as a, b, and c) with
the convention of side a being opposite from vertex A, side b being opposite from vertex B,
and so forth.
B
c
a
A
b
C
We will use the vertex name (e.g., A) to refer to both the vertex and the angle at that vertex,
trusting that context will make clear which is appropriate.
Here are several important facts about triangles:
– In flat space (i.e., Euclidean space such as the Cartesian coordinate plane with which
you are familiar), the sum of all three angles in any triangle must add up to 180◦ .
Note: This is not true in non-flat (i.e., non-Euclidean) space. For instance, on the surface of
a sphere the angles in a triangle add up to more than 180◦ . Take as an example a particular
triangle on the surface of the Earth. Imagine you start out on the Equator
and travel due north to the North Pole. Once there, if you take a 90◦ right
turn and head due south to the Equator, finally turning right 90◦ to follow the
Equator west back to your starting point, you will have traced out a triangle.
The angles in this triangle, however, add up to 270◦ . Don’t worry; from here
on out, we’ll restrict our discussion to flat space.
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TRIGONOMETRY
2 SIMILAR TRIANGLES
– All triangles abide by what is known as the triangle inequality, which is actually the
following three inequalities.
a+b>c
b+c>a
a+c>b
Essentially, these inequalities say that the sum of lengths of any two sides of a triangle must
be greater than the length of the remaining side. Try to construct a triangle that does not
satisfy one of the inequalities (say where a + c < b, for instance), and you’ll quickly see why
the triangle inequality must be true.
a
c
b
If two of the sides together aren’t longer than the third, then you don’t have a triangle. Why
is it not enough for the sum of two side lengths to be equal to the third side length?
– A right triangle is a triangle with one right (i.e., 90◦ ) angle. The side opposite the
right angle is called the hypotenuse and the other two sides are called legs.
U
hypotenuse
leg
T
V
leg
Can a triangle (in flat space) have more than one right angle? Why or why not? What can
we say about the sum of the measures of angles U and V above?
2.
Similar Triangles
Now we introduce a concept central to the rest of our discussion: similarity. Where congruent triangles are essentially identical (up to rotation and reflection), similar triangles
merely have the same shape (identical up to rotation, reflection, and dilation). That is, with
two congruent triangles, you can move one over to the other, rotate it in the correct way,
perhaps flip it over, and they will line up exactly. A pair of similar triangles, however, only
share the same basic shape and may differ in size. Consider the following, in which triangle ABC is congruent to triangle DEF (4ABC ∼
= 4DEF ) but similar to triangle P QR
(4ABC ∼ 4P QR).
Q
B
c
A
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b
E
f
C
D
r
p
d
e
F
2
P
q
R
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TRIGONOMETRY
2 SIMILAR TRIANGLES
It should be clear that congruence is a stronger condition than similarity. In other words,
it is easier for two triangles to be similar (have the same shape) than it is for them to be
congruent (have the same shape and size). What, then, is the defining quality of a pair of
similar triangles? If the two aren’t required to be the same size, then the corresponding side
lengths need not be identical, as they must for a pair of congruent triangles. Examine the
following, in which 4T U V ∼ 4XY Z ∼ 4ABC.
A
X
z
T
v
b
y
u
V
c
t
Y
U
x
Z
B
a
C
For triangles to be similar, their corresponding angles must be equal in measure and their
corresponding sides must be proportional. That is, in the example of 4T U V and 4ABC
above, angle T must have the same measure as angle A, U the same as B, and V the same
as C. In addition, the ratio of side length t to side length a must be equal to the ratios of
the other two pairs of corresponding sides:
u
v
t
= = .
a
b
c
As it turns out, for triangles it is enough to guarantee similarity if all three pairs of
corresponding angles are congruent; the proportionality of corresponding side lengths follows
automatically. (Note: This isn’t necessarily the case for other polygons. Otherwise all
rectangles would be similar, which is clearly not true.) This is a useful (and important) fact
since it allows us to conclude that triangles are similar if just two pairs of corresponding
angles have the same measure. Why is that?
There is another important fact about similar triangles. To see it, let us consider two of
the similar triangles above, 4ABC and 4XY Z. The proportionality of corresponding side
lengths tells us that
a
c
= .
x
z
With a bit of algebra (multiplying both sides by x and z), we find that
az = cx ,
and further (dividing both sides by c and z), that
a
x
= .
c
z
Thus if two triangles are similar, the corresponding ratios of side lengths are identical as
well. That is,
a
x
a
x
b
y
= ,
= ,
= , etc.
b
y
c
z
c
z
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TRIGONOMETRY
2 SIMILAR TRIANGLES
Let us be clear about the difference between these two statements concerning the proportionality of side lengths. The first statement, from our definition of similarity for triangles,
says that the corresponding sides of two similar triangles must all be proportional. That is,
the ratio created by dividing a side length of one triangle by the length of its corresponding
side from the other triangle is the always the same, no matter which corresponding sides you
choose.
b
a
d
a b
=
d e
e
Ratios of corresponding sides are equal
The second statement, which we found after some algebra, says that the corresponding ratios
of sides from within each triangle are equal. That is, the ratio created by dividing two side
lengths from the same triangle is the same as the ratio of the corresponding two sides from
the other triangle, no matter which ratio you choose.
b
a
d
a d
=
e
b
e
Corresponding ratios of sides are equal
The distinction is subtle but important. It is this second statement about the corresponding
ratios of side lengths that will be the foundation for what we call the trigonometric functions.
Consider the following similar triangles, where the measures of angles A, A0 , A00 , and A000
are all equal.
a0
c
a
A000
a00
c0
c 000
A
a000
c 00
A0
A00
From our above result, we know that
a
a0
a00
a000
= 0 = 00 = 000 .
c
c
c
c
The ratios are identical for all four triangles even though the side lengths are not. If the
ratio were different for one of the triangles, we would know one of two things. Either the
corresponding sides have been chosen incorrectly, or that triangle is not similar to the others.
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TRIGONOMETRY
3 THE SINE FUNCTION
In both cases, this would mean that the angle in question is different (with the one possible
exception that our differing ratio is upside down). Therefore we can think of the ratio of side
lengths as dependent on the angle: if the angle is the same, the ratio will match; if the angle
is different, then so too will be the ratio. This result holds for any class of similar triangles,
but let us deal exclusively with right triangles from here on out; they’re nicer for a variety
of reasons.
3.
The Sine Function
Consider the following right triangle with non-right angle θ.
hyp
opp
θ
adj
In the figure, the side labeled “opp” is opposite the angle θ in question, the side “adj” is
adjacent to angle θ, and “hyp” is the hypotenuse of the triangle, opposite the 90◦ angle. For
right triangles, let us define the sine function on the angle θ by
sin(θ) =
opp
.
hyp
Any right triangle with an angle equal in measure to θ must be similar to the above triangle
(since having any two congruent corresponding angles guarantees all three will be congruent).
Thus, for any right triangle with the same angle θ, while its corresponding side lengths may
be quite different depending on its size, its ratio opp
will always be the same. For a different
hyp
.
angle θ, the triangle will not be similar and may therefore have a different ratio opp
hyp
As an example, consider the following triangle.
2
30◦
√
60◦ 1
3
◦
◦
◦
We know from
√ basic geometry that all 30 -60 -90 triangles have side lengths which are in
the ratio 1 : 3 : 2. Thus we find that
sin(30◦ ) =
and
1
2
√
3
sin(60 ) =
.
2
◦
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TRIGONOMETRY
4 CIRCLES AND ANGLE MEASURE
These results hold even if we consider a different 30◦ -60◦ -90◦ triangle, say one twice as
large.
60◦
4
2
30◦
√
2 3
Again we find that
1
2
=
4
2
√
√
2 3
3
◦
sin(60 ) =
=
.
4
2
sin(30◦ ) =
and
What is sin(45◦ )?
So long as the angle in question has the correct measure, it doesn’t matter which right
triangle you use to determine the ratio. Any similar triangle will have the same ratio and
therefore give the same value for sin(θ). Nor does it matter what units you use to measure
the angle; the ratio of side lengths will be the same. Thus we see that the sine function is
well-defined for any angle θ where 0◦ < θ < 90◦ . We will extend the domain of this function
to all real numbers momentarily.
4.
A Note On Circles and Angle Measure
There are several common units used to measure angles: revolutions, degrees, and radians. The most intuitive of these is the revolution. The idea of 2.5 revolutions (two and a
half turns around) is much more accessible than 900◦ or 5π radians. Due to exposure, you
are probably most familiar with degree measure and likely have a clear picture in your mind
of what a 45◦ angle looks like. A 1.3-radian angle is certainly less clear. While the three
units may seem very different, they all measure the same thing. In fact, we can convert
between them because we know that
1 rev = 360◦ = 2π rad .
How many degrees does a 1-radian angle span?
The constant π is ubiquitous throughout mathematics and the mathematical sciences.
You have likely seen it before in the familiar formulas for the circumference and area of circles
(C = 2πr and A = πr2 , respectively), but may be unclear as to what exactly it means. It is
simply the ratio of a circle’s circumference (the distance around its outside) to its diameter
(its width).
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TRIGONOMETRY
4 CIRCLES AND ANGLE MEASURE
C
r
π=
C
D
D
It turns out that the value of π is irrational (meaning that it cannot be expressed as a ratio
of two integers) and thus has an infinite, non-repeating decimal representation:
π = 3.141592 . . .
For any circle, no matter how large or small, this ratio will always hold.
Note: It is the result of unfortunate historical precedent that we give π so much attention. The better ratio to consider would be that of a circle’s circumference to its radius,
since the radius is actually the more fundamental measurement for a circle. What is a circle,
after all? It’s the collection of points some fixed distance from one special point (the center)
where the distance in question is of course the radius. Being precise about even the definition of diameter is a bit clumsy, whereas the definition of radius is crystal clear. Because
radius is half of the diameter, the ratio Cr holds for all circles as well, and is in fact twice
the numerical value of π. If the common constant was the ratio of circumference to radius
(instead of to diameter), many things in mathematics and science would seem more natural
and be more consistent. One good example of this is the unit circle, which we will discuss
shortly. Do a web search for “tau manifesto” if you are interested in reading more about our
choice of circle constant.
If all of the units we use for angle measure are equivalent, then what difference does it
make which we use? To see, consider trying to determine arc length, the length of some
portion of a circle.
L
θ
r
It should be clear that the ratio of the arc length L to the circumference of the circle must
equal the ratio of the angle θ to the angle representing going all of the way around the circle.
That is,
L
θ
=
.
circumference
entire circle
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TRIGONOMETRY
4 CIRCLES AND ANGLE MEASURE
Using this proportionality, we can derive an equation that gives the length of an arc L in
terms of the radius r of the circle on which it lies and the corresponding angle θ swept out.
However, this equation will be different depending on the units we use for angle measure.
Revolutions
θ
L
=
2πr
1 rev
Degrees
Radians
θ
L
=
2πr
360◦
θ
L
=
2πr
2π
2πrθ
1 rev
L=
2πrθ
360◦
L = 2π rθ
L=
π
rθ
180◦
L=
L=
2πrθ
2π
L = rθ
This is the first example where we see that, in some sense, radians are a more natural unit
for angle measure. From now on, we take radians as our unit of choice. Unless revolutions
are explicitly mentioned or a degree symbol (◦ ) is used, we will assume angle measures are
in radians.
We found that measuring angles in radians has the advantage of leading to a simple
equation relating radius and arc length, namely L = rθ. Examining this equation further
helps to give us a geometric understanding of what exactly a radian is. If we consider a
1-radian angle (θ = 1), then our equation tells us that
L=r
L=r·1
=r.
1
r
That is, one radian is the angle that must be swept out for the arc length to equal the radius
of the circle.
A very useful tool for studying trigonometric functions like sine is the unit circle. The
unit circle is a circle of radius 1 (in whatever units you are using), typically centered at the
origin. It is given by the equation x2 + y 2 = 1.
y
1
1 x
−1
−1
Do you see why, for arcs on the unit circle with angles measured in radians, L = θ?
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TRIGONOMETRY
5.
5 TRIG. FUNCTIONS AND THEIR GRAPHS
The Basic Trigonometric Functions and Their Graphs
For any right triangle with angle θ,
hyp
opp
θ
adj
we define the three basic trigonometric functions as follows.
Cosine
Sine
sin(θ) =
opp
hyp
cos(θ) =
Tangent
adj
hyp
tan(θ) =
opp
adj
4
◦
20
40 ◦
,2
4π
3 ,2
5π
7π
4
◦ , 4
5 5π
31 0◦ , 3
30
25 ◦
◦
35
,1
4
60 ◦
45 ◦ , 3π
, π
2π , 1
3
3π
sin(θ)
Can you show that tan(θ) = cos(θ)
?
We have spent a considerable amount of time preparing the foundation for and defining
the basic trigonometric functions, but how does this relate to waves and cyclic phenomena?
To answer that question, we must consider the graphs of these functions. In order to determine what the graphs of the trigonometric functions look like, we need to consider the unit
circle.
First, we will adopt the convention of measuring our angles counter-clockwise from the
positive x-axis. That is, for the point (1, 0) on the unit circle, the corresponding angle θ is
zero. Moving counter-clockwise from that
y
location gives points whose corresponding angles are positive. Moving clockwise
π
◦
2 90
from that location gives points whose corresponding angles are negative. In the
figure at right, some common angles are
5π
π
◦
shown, with the degree measures toward
6 ,1
+
50 ◦
0, 6
3
the inside of the circle and the radian measures toward the outside. Of course, we
π, 180◦
0◦ , 0
x
are dealing with a circle, so every point
360◦ , 2π
has infinitely many corresponding angles.
330 ◦
◦
0
Name three angles that all correspond to
1
, 11π
2
7π ,
6
the point (0, −1).
6
3π
2
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TRIGONOMETRY
5 TRIG. FUNCTIONS AND THEIR GRAPHS
Consider taking a point on the first quadrant of the unit circle as one vertex of a right
triangle (along with the point on the x-axis directly below it together with the origin).
1
(x, y) = (cos θ, sin θ)
y
θ
x
1
Because the hypotenuse has a length of one, we find that sin(θ) is equal to the y-coordinate
of our point and cos(θ) is equal to its x-coordinate. Be sure that you see why this is the
case.
Note that when it is clear what argument is being given to a trigonometric function, we
may leave off the parentheses, so sin θ = sin(θ). We keep the parentheses if there might
be any confusion about what is being plugged into the function, so cos(θ + 1) rather than
cos θ + 1 (unless of course what you intend is indeed cos(θ) + 1).
With the above figure in mind, it becomes clear how we can define our trigonometric
functions for all angles θ (not just those in between 0 and π2 radians). Starting with the
point (1, 0) (i.e., the one corresponding to θ = 0), and following it around as θ increases (or
decreases), we see that the value of the function sin θ is just the y-coordinate of our point
on the unit circle. Since wrapping multiple times around the circle can get us back to the
same point, we expect the sine function to be periodic.
y
x
Examining the graph of the function below, we see that, as expected, sine is zero when
θ is zero, it increases up to a maximum of one (where θ = π2 at the point (0,1)), it decreases
down to a minimum of negative one (where θ = 3π
at the point (0, −1)), and it begins all
2
over again at zero when θ = 2π.
sin θ
1
− π2
π
π
2
3π
2
2π
θ
−1
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TRIGONOMETRY
5 TRIG. FUNCTIONS AND THEIR GRAPHS
Similarly, we see that as our point traces around the unit circle, its x-coordinate gives us
the value of the cosine function, whose graph is shown below.
cos θ
1
− π2
π
π
2
3π
2
θ
2π
−1
It will be a worthwhile exercise for you to stop and determine the x- and y-coordinates of all
of the points corresponding to the common angles given on the unit circle back on page 9.
In the context of our point traveling around the unit circle, the tangent function is given
by tan θ = xy . (Be sure that you understand why this is so.) Because the x-coordinate is
sometimes zero, this means that there will be values of θ for which tangent is undefined.
In fact, because the sine and cosine values are not both going to zero at the same time,
the values of θ near where cosine goes to zero cause the values of tan θ to grow unbounded,
producing vertical asymptotes. Consider the following graph of the tangent function.
tan θ
− π2
π
2
π
3π
2
θ
Let us discuss one final bit of notation you will encounter. Considering the expression
sin θ·sin θ, one might expect it to be written as sin θ2 . Unfortunately, this could be interpreted
as meaning either of the following: sin(θ2 ) or (sin θ)2 . To avoid this ambiguity, when we mean
the former, we always write sin(θ2 ), and we adopt the somewhat shorter notation for the
latter of sin θ · sin θ = (sin θ)2 = sin2 θ (often read as “sine squared of theta”).
Can you prove that sin2 θ + cos2 θ = 1 for all values of θ?
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