124 Chapter 4 Let’s Learn Integration Techniques!
Incidentay, since the
x-axis coincides with
the y-axis when it is
rotated by 90degrs
(
π
2
radians), we can
say
sin
θ
is a function
that outputs, delayed
by
π
2
, the same values
as
cos
θ
.
In other words,
sin cos
θ
π
θ
+
=
2
Yes?
Uh…wi you give
us back our
drumsticks?
Now, we are ready
for the main part of
the Sanda Suer
Festival!!
y
θ
x
cos θ
sin (θ + )
2
θ
Oops!
Using Integrals with Trigonometric Functions 125
Here are special
seats for you. Be
careful not to
fa, reporters,
and take gd
pictures.
Okay. We wi.
Now, we are
going to lk at
cos
θ
in terms of
calculus!
Mr. Seki,
your actions
are totay
dierent from
what you say.
In fact, integrals are
easier to obtain than
derivatives.
It’s easier to
understand if
we lk down
at the circle of
dancers from
way up here.
Using Integrals with Trigonometric Functions
126 Chapter 4 Let’s Learn Integration Techniques!
What we nd to do is to find out what
∑ cos
θ
× Δ
θ
= cos
θ
0
(
θ
1
−
θ
0
) + cos
θ
1
(
θ
2
–
θ
1
) + …
+ cos
θ
n−1
(
θ
n
–
θ
n−1
) becomes.
Lking at this
puts me in a
fog.
Lk at this
figure. Doesn’t
this give you a
gd idea? This
shows that the
intersecting
angleof the
y-axis with the
tangent line
PQ,
where P is the
point moved from
(1, 0) by angle
θ
,
isalso
θ
.
?
Futoshi! Why does
he get to eat chow
mein while I have
to learn about
integrals?
1
−1
2
3
2
5
2
2
0
y
P
1
θ
x
Q
θ
0 1
y
A′
1
x
0
A
1
A′
2
A
2
A
0
A
3
θ
2
− θ
1
θ
3
− θ
2
θ
1
− θ
0
At angle θ
1
with the y-axis
Length θ
2
− θ
1
The change in cos θ is the length A′
1
A′
2
.
That length is the orthogonal projection A
1
A
2
.
Length A′
1
A′
2
≈ arc A
1
A
2
× cos θ
1
= (θ
2
− θ
1
) × cos θ
1
Chow Mein
Using Integrals with Trigonometric Functions 127
Let’s use this
to integrate
from 0 to
α
.
Uh…
Is this right?
Right! If we
make these
infinitely
sma…
We find that
the integral
of cosine is
sine.
Then, to put it the
other way around, the
derivative of sine is
cosine?
You’re
right!
Now,
remember
these
formulas.
A
1
(cos θ
1
, sin θ
1
)
A
2
A
n
(cos θ
n
, sin θ
n
) = (cos α, sin α)sin α = sin θ
n
A
0
(cos θ
0
, sin θ
0
) = (1, 0)
α
y
A′
1
x
0
A′
2
∑ cos
θ∆θ
when
θ
is changed from 0 to
α
cos θ
0
(θ
1
− θ
0
) + cos θ
1
(θ
2
− θ
1
) + … + cos θ
n−1
(θ
n
− θ
n−1
)
≈ A′
0
A′
1
+ A′
1
A′
2
+ … + A′
n−1
A′
n
= A′
0
A′
n
= sin α
128 Chapter 4 Let’s Learn Integration Techniques!
A right! Let’s
do the calculus
dance song!!
Calc…
That’s a
strange
sound!
Formula 4-1: The Dierentiation and Integration of Trigonometric Functions
Since u
cos sin sin
θ θ α
α
d = −
∫
0
0
, we know that sine must be cosine’s derivative.
v
sin cos
θ θ
( )
′
=
Now, substitute
θ
+
π
2
for
θ
in v. We get sin cos
θ
π
θ
π
+
′
= +
2 2
.
Using the equations from page 124,
we then know that
w
cos sin
θ θ
( )
′
= −
We find that differentiating or integrating sine gives cosine and vice versa.
Calc
Calculus Dance Song
Trigonometric version
Raise
both arms
toward
uer right.
Jump and
turn to
the left.
Jump again
and clap
your hands
twice.
Calc
Calculus
Calc
Calculus
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