Page 92 - Numerical Analysis Using MATLAB and Excel

Characteristics of Sinusoids

i = 6sin ( 100πt – 210 ) ° = 6sin ( 100πt + 150 ) °
2
°
= 6cos ( 100πt + 150 – 90 ) ° = 6cos ( 100πt + 60 ) °

and comparing with , we see that leads by 90° , or lags by 90° .
i
i
i
i
i
i
2
1
1
2
2
1
In our subsequent discussion, we will be using several trigonometric identities, derivatives and
integrals involving trigonometric functions. We, therefore, provide the following relations and
formulas for quick reference. Let us also review the definition of a radian and its relationship to
degrees with the aid of Figure 3.5.
1 radian = 57.3 deg
r
r

π radians

Figure 3.5. Definition of radian

As shown in Figure 3.5, the radian is a circular angle subtended by an arc equal in length to the
r
radius of the circle, whose radius is units in length. The circumference of a circle is 2πr units;
therefore, there are 2π or 6.283… radians in 360° degrees. Then,

360°
1 radian = ----------- ≈ 57.3° (3.3)
2π
The angular velocity is expressed in radians per second, and it is denoted by the symbol . Then,
ω
n
a rotating vector that completes revolutions per second, has an angular velocity ω = 2πn radi-
ans per second.

Some useful trigonometric relations are given below for quick reference.

cos 0° = cos 360° = cos 2π = 1 (3.4)

π 3
cos 30° = cos --- = ------- = 0.866 (3.5)
6 2

π 2
cos 45° = cos --- = ------- = 0.707 (3.6)
4 2
π 1
cos 60° = cos --- = -- = 0.5 (3.7)
-
3 2

Numerical Analysis Using MATLAB® and Excel®, Third Edition 3−5
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