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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