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Derivative of the Sine Function
By James Nickel, B.A., B.Th., B.Miss., M.A.www.biblicalchristianworldview.net
Basic Trigonometry
• Given the unit circle (radius = 1) with angle (theta), then, by definition, PA = sin and OA = cos . 1
P
O cos
sin
A
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Basic Trigonometry
• As changes from 0 to /2 radians (90), sin also changes from 0 to 1.
• As changes from 0 to /2 radians, cos also changes from 1 to 0.
1
P
O cos
sin
A
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Question
• How does sin vary as varies?
• To find out, we must calculate the derivative of y = sin.
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The Method of Increments
• We add to and determine what happens.
• We need to find the change in y or y.
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1
O
A
P
sin
cos
y
The Method of Increments
• PA has increased to QB where QB = sin( + ).
• OA has decreased to OB where OB = cos( + ).
also represents the radian measure of the arc from P to Q.
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1
O
A
P
Q
B
1
The Method of Increments
• Also remember that is just a “little bit” or infinitesimally small.
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1
O
A
P
Q
B
1
Our Goal
• Find QD = y• To do this, we must
show QDP ~ PAO• We then let 0• Hence,
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y DQ OAPO
1
O
A
P
Q
B
1
D
Some Geometry
• Let’s now draw the tangent line l to the circle at point P.
• We know that l PO (the tangent line intersects the circle’s radius at a right angle).
• Hence, QPO = 90
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1
O
A
P
Q
B
1l
D
Some Geometry
• Construct DP OA• Note the transversal OP• Hence, = DPO QPO = 90 DPO and
DPQ are complementary.• Hence, DPQ = 90 –
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1
O
A
P
Q
B
1l
D
sin
Some Geometry
DPQ = 90 – DQP =
• Hence, QDP ~ PAO because both are right triangles and DQP = POA =
Copyright 2009www.biblicalchristianworldview.net
1
O
A
P
Q
B
1l
D
Some Geometry
• Consider DQ• It represents the change
in y (i.e., y) resulting from the change in (i.e., ).
sin siny DQ
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1
O
A
P
Q
B
1l
D
sin
Derivative Formula
sin siny DQ
In QDP, cos = DQ
Note that as gets infinitesimally small, QP (the hypotenuse) converges to .
Copyright 2009www.biblicalchristianworldview.net
1
O
A
P
Q
B
1l
D
sin
Derivative Formula
• Because QDP ~ PAO, then:
DQ OAPO
Copyright 2009www.biblicalchristianworldview.net
1
O
A
P
Q
B
1l
D
sin
Derivative Formula• Since PO = 1 and OA =
cos , then:
y DQ
sin sin
OA coscos
PO 1
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1
O
A
P
Q
B
1
l
D
sin
OA=cos
Derivative Formula
• As we let approach 0 as a limit, then: y (sin) = cos
• This means as increases, sin increases at a instantaneous rate of cos.
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Derivative Graphy-axis
-axis
y = sin
• The solid line graph represents y = sin (the sine curve).
• The dotted line graph represents y = cos (the cosine curve).
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Derivative Graphy-axis
-axis
y = sin
• Note that when = 0, then cos = 1. • This means that the slope of the line tangent to
the sine curve at 0 is 1. • The cosine curve plots that derivative.
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Derivative Graphy-axis
-axis
y = sin
• When = /2, then cos = 0. • This means that the slope of the line tangent to
the sine curve at /2 is 0 (the slope is parallel to the -axis).
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Derivative Graphy-axis
-axis
y = sin
• The cosine curve traces the derivative of the sine curve at every point on the sine curve.
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