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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