Projects - Cengage Learning

Appendix C C-47
PROJECT
A Linear Approximation for the Sine Function
In this project we derive the following very important property of the sine
function:
sin u
For real numbers u, the ratio approaches 1 as u approaches zero.
u
In the phrase “as u approaches zero” (in the domain of sin u) it is important to
understand that u cannot be zero because we can’t substitute zero for u in our
trigonometric ratio.
To begin we need a preliminary result:
For real numbers u, cos u approaches 1 as u approaches zero.
Exercise 1 Convince yourself that this preliminary result is true in each of
the following ways.
(a) By looking at the graph of y cos u, for p4 u p4, in a u-y coordinate
system.
(b) By looking at a point (cos u, sin u) on the unit circle.
In both parts (a) and (b), be sure to consider u approaching zero through both
positive and negative values.
Now consider real numbers u approaching zero through positive values.
Eventually u p2 so we may assume 0 u p2. In Figure A we have the
unit circle with various labeled points:
O is the origin.
A is the point (1, 0).
P is the point on the unit circle corresponding to the number u, or,
equivalently, the angle with radian measure u.
B is the point at the foot of the perpendicular from P to the x-axis.
Q is the vertex of ^OAQ lying on the terminal side of angle u outside
the unit circle; the line through Q and A is tangent to the circle.
y
Q
O
¨
P
B
A
x
Q(1, tan ¨)
P(cos ¨, sin ¨)
¨
Figure A
≈+¥=1
¨
O B(cos ¨, 0)
Figure B
A(1, 0)
Exercise 2 In Figure A explain why OAQ is a right angle. Then explain why
P (cos u, sin u), B (cos u, 0), and Q (1, tan u)
The information from Exercise 2 is shown in Figure B.