Download Lecture 1 - University of Chicago Math

Lecture 1
Math 13200
January 4
Recall the definitions of sin, cos, tan in terms of the mnemonic sohcahtoa. sohcahtoa says that given
a right triangle ABC with side lengths x, y, z (with z the length of the hypotenuse), we have
B
sin θ =
opp
y
=
hyp
z
cos θ =
adj
x
=
hyp
z
tan θ =
opp
y
=
adj
x
z
y
θ
C
x
A
Implicit in these definitions of the trigonometric functions is that the ratio of two sides does not
depend on the right triangle, but only the angle θ. Indeed, this follows from the angle-angle theorem
of high school geometry. It states that if two triangles have two equal angle measures, then they
are similar. Thus, the ratios of corresponding sides are equal.
First note that sin and cos are never greater than 1. One can use either the Pythagorean Theorem,
(which says x2 +y 2 = z 2 ) or use the triangle inequality (which says that x+y ≤ z) to see this.
The above discussion implies that any trigonometric function, say sin, gives a function
sin : {angles θ with 0◦ < θ < 90◦ } → {ratios (or percentages)}
(Recall our notation f : X → Y which states that f is a function with domain X and target
Y .)
By thinking of angles and ratios as numbers, this gives us a real function of one real variable
sin : (0, 90) → [0, 1]
These are the types of functions we looked at last quarter! If you plug an x-value (between 0 and
90) in, and get a y-value out. Further, if you remember what the graph of sin looks like (i.e.,
smooth), you should be convinced that we should be able to do calculus with it.
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Unfortunately, the sohcahtoa definition is insufficient for our purposes. It is great for computing the
measurements of angles and sides of triangles. However there are some drawbacks of only thinking
of trigonometric functions only in terms of right triangles and ratios of sides:
1. What about the angles 0◦ and 90◦ ? Our intuition about geometry and our knowledge of limits
from last quarter lead us to believe
• sin 0 = 0
• sin 90 = 1
• cos 0 = 1
• cos 90 = 0
• tan 0 = 0
• tan 90 = +∞ (on the left side - is there even a right side?)
But what does this even mean? There doesn’t exist a right triangle with a side length of 0.
2. Even more - there are plenty of other angles other than the ones between 0 and 90 - as
any BMX competition will readily show. What is sin 293487◦ ? There does not exist a right
triangle with one of its angles equal to 293487◦ - so sohcahtoa tells us nothing.
3. The relationship between the graphs of trig functions and sohcahtoa is extremely limited.
This is unfortunate, because studying functions is made easier by understanding their graphs.
4. Our rigorous definitions of continuity and differentiation are awkward to work with in terms
of right triangles/ratios of sides.
5. The sum of angles identity is very difficult to prove using the definition in terms of right
triangles/ratios of sides.
6. Angles are key for the measurement and geometry of triangles. But angles are especially
related to the geometry of circles and circular arcs. For example, every point of a circle can
be uniquely distinguished by an angle in [0, 360).
We should ask - does this imply a relationship between trig functions and circles? In fact
yes.(Indeed - the origin of trigonometry comes from the study of stars as they traced circular arcs in the celestial sphere.) The sohcahtoa definitions don’t directly reflect this deep
relationship between circles and trigonometric functions.
So we will define trigonometric functions differently. We will use the last point from above about
circles to motivate our definition.
WARNING: From this point on, we shift from degrees to radians.
Consider the unit circle C in the x-y plane - that is, the circle of radius 1 centered at O = (0, 0).
Equivalently, the set of all points (x, y) such that x2 + y 2 = 1. Let A be the point (1, 0) on C. For
each number θ ≥ 0 (respectively ≤ 0), trace counterclockwise (respectively clockwise) around the
circle an arc length of θ (in terms of radians, this point corresponds to θrad). Let the endpoint of
this arc be Pθ = (xθ , yθ ). We say Pθ is the point on C corresponding to the angle θ.
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For θ a real number, we define sin θ to be the y-value of the point Pθ on C corresponding to the
angle θ. Similarly, we define cos θ to be the x-value of Pθ . Finally, we define tan θ to be the slope
of the line through O and Pθ .
y
P
sin θ
O
θ
θ
cos θ
A
x
We immediately deduce the following two trigonometric identities:
sin2 θ + cos2 θ = 1
tan θ =
sin θ
cos θ
The first comes from the fact that P = (cos θ, sin θ) lies on the unit circle (which is defined by the
condition that x2 + y 2 = 1). The second follows from the slope formula
tan θ =
sin θ
sin θ − 0
=
cos θ − 0
cos θ
We can additionally deduce some other trig identities from the geometry of the situation (these
don’t even make sense in the sohcahtoa setting):
sin(θ + 2π) = sin(θ)
cos(θ + 2π) = cos(θ)
tan(θ + 2π) = tan(θ)
sin(−θ) = − sin(θ)
cos(−θ) = cos(θ)
tan(−θ) = − tan(θ)
(In homework you will be asked to prove these. This will come down to understanding the relationships between Pθ , P−θ , and Pθ+2π .)
Also, it is not so difficult to see that this definition gives the same definition as sohcahtoa. This is
because the hypotenuse is equal to 1 - so the ratio y/z = opp/hyp is equal to y/1 = y (similarly
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for x and adj. Since the sohcahtoa definition of the trig functions we gave at the beginning only
depend on the angle (i.e., we can pick any right triangle containing the angle θ to calculate the
trig functions), we choose the one inscribed in the unit circle. This implies that our new definition
continues to satisfy the sohcahtoa identities.
Does our new definition solve the problems that we laid out at the before introducing it? It clearly
solves the first two problems. Number 4 is also answered affirmatively as we will see in future
lectures. Number 5 is a homework problem. Number 6 is answered with a resounding yes - indeed,
the complexity of the trigonometric functions reflects the clash of imposing a circle on a rectangular
coordinate system.
How about number 3 - how do we see the graph of sin from our new geometric picture? First
imagine that θ is a time variable, and a point P moves around the circle counterclockwise at a
constant speed of 1. The y-value of the point at time θ is (by definition) sin θ. So to understand
the graph of sin, we just move around the circle, while only paying attention to the y-value.
From time 0 to π/2, the point’s y-value increases from 0 to 1, while its speed decreases to 0 once it
hits π/2. From π/2 to 3π/2, the y-value decreases from 1 to −1, hitting 0 at π. From 3π/2 to 2π,
it returns from −1 back to 0. Clearly, this same back and forth pattern occurs on every interval of
the form [2πn, 2π(n + 1)]. We see this reflected in the graph of sin:
Graph of sine.
Similarly, we can consider tangent...
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