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Derivatives of the Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018
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• Derivatives of the Trigonometric FunctionsMATH 161 Calculus I

J. Robert Buchanan

Department of Mathematics

Summer 2018

• Background

We can establish formally the limits of the trigonometricfunctions using circles, angles, and geometry.

Recall:I The length s of the arc of a circle of radius r which

subtends a central angle is s = r .I The area A of a sector of a circle of radius r subtended by

a central angle is A = 12 r2.

I The coordinates of the point on the circle of radius rcentered at the origin at the intersection of a ray making anangle with the positive x-axis are

(x , y) = (r cos , r sin ).

• Limits and TrigonometryWe can establish formally the limits of the trigonometricfunctions using the unit circle and geometry.

sin (cos ,sin )

x

y

• Basic Limits (1 of 2)

Lemmalim0

sin = 0.

Proof.If 0 < < 2 then

0 sin lim0

0 lim0

sin lim0

0 lim0

sin 0

and use the Squeeze Theorem.

• Basic Limits (2 of 2)

Lemmalim0

cos = 1.

Proof.If 0 < < 2 then

lim0

cos = lim0

1 sin2

=

1

(lim0

sin

)2=

1 02 = 1.

• Justification of a Common Limit (1 of 3)

R(1,0)O

P(cos ,sin )

Q(1,tan )

x

y

4OPR = 12

(1) sin

4OQR = 12

(1) tan

OPR =12

(12)

• Justification of a Common Limit (2 of 3)

Lemma

lim0

sin

= 1

• Justification of a Common Limit (3 of 3)

Proof.

0 < area OPR < area OPR < area OQR

0