Basic Stuff
1.1 Trigonometry
The common trigonometric functions are familiar to you, but do you know some of the tricks to
remember (or to derive quickly) the common identities among them? Given the sine of an angle, what
is its tangent? Given its tangent, what is its cosine? All of these simple but occasionally useful relations
can be derived in about two seconds if you understand the idea behind one picture. Suppose for example
that you know the tangent of
θ
, what is
sin
θ
? Draw a right triangle and designate the tangent of
θ
as
x
, so you can draw a triangle with
tan
θ
=
x/
1
.
1
θ
x
The Pythagorean theorem says that the third side is
√
1 +
x
2
. You now
read the sine from the triangle as
x/
√
1 +
x
2
, so
sin
θ
=
tan
θ
√
1 + tan
2
θ
Any other such relation is done the same way. You know the cosine, so what’s the cotangent? Draw a
different triangle where the cosine is
x/
1
.
Radians
When you take the sine or cosine of an angle, what units do you use? Degrees? Radians? Cycles? And
who invented radians? Why is this the unit you see so often in calculus texts? That there are
360
◦
in
a circle is something that you can blame on the Sumerians, but where did this other unit come from?
R
2
R
s
θ
2
θ
It results from one figure and the relation between the radius of the circle, the angle drawn,
and the length of the arc shown. If you remember the equation
s
=
Rθ
, does that mean that for a
full circle
θ
= 360
◦
so
s
= 360
R
? No. For some reason this equation is valid only in radians. The
reasoning comes down to a couple of observations. You can see from the drawing that
s
is proportional
to
θ
— double
θ
and you double
s
. The same observation holds about the relation between
s
and
R
,
a direct proportionality. Put these together in a single equation and you can conclude that
s
=
CR θ
where
C
is some constant of proportionality. Now what is
C
?
You know that the whole circumference of the circle is
2
πR
, so if
θ
= 360
◦
, then
2
πR
=
CR
360
◦
,
and
C
=
π
180
degree

1
It has to have these units so that the left side,
s
, comes out as a length when the degree units
cancel. This is an awkward equation to work with, and it becomes
very
awkward when you try to do
calculus. An increment of one in
Δ
θ
is big if you’re in radians, and small if you’re in degrees, so it
should be no surprise that
Δ sin
θ/
Δ
θ
is much smaller in the latter units:
d
dθ
sin
θ
=
π
180
cos
θ
in degrees
James Nearing, University of Miami
1
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1—Basic Stuff
2
This is the reason that the radian was invented. The radian is the unit designed so that the propor
tionality constant is one.
C
= 1
radian

1
then
s
=
(
1
radian

1
)
Rθ
In practice, no one ever writes it this way.
It’s the custom simply to omit the
C
and to say that
s
=
Rθ
with
θ
restricted to radians — it saves a lot of writing. How big is a radian? A full circle has
circumference
2
πR
, and this equals
Rθ
when you’ve taken
C
to be one. It says that the angle for a
full circle has
2
π
radians. One radian is then
360
/
2
π
degrees, a bit under
60
◦
. Why do you always use
radians in calculus? Only in this unit do you get simple relations for derivatives and integrals of the
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 Fall '08
 Nearing
 Pythagorean Theorem, Lebesgue, basic stuff, Handbook of Mathematical Functions

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