PROPERTIES OF SINE AND COSINE FUNCTIONS

PROPERTIES OF SINE AND COSINE FUNCTIONS:

1. The sine and cosine functions are both periodic with period 2π.
2. The sine function is odd function since it’s graph is symmetric with respect to the origin, while the cosine function is an even function since it’s graph is symmetric with respect to y axis.
3. The sine functions:
a. Increasing in the intervals[0, π/2]and [3π/2, 2π]; and
b. Decreasing in the interval [π/2, 3π/2],over a period of 2 π.
4. The cosine function is:
a. Increasing in the interval [π, 2π]; and
b. Decreasing in the interval [0, π], over a period 2π.
5. Both the sine and cosine functions are continuous functions.
6. The domain of the sine and cosine functions is the set of all real numbers from -1 to 1
7. The amplitude of both the sine and cosine functions is 1, since one-half of the sum of the lower bound is 1, that is ½[|1|]+[|-1|]=2/2 or 1.
8. The maximum and minimum values of the sine and cosine functions are 1 and -1 respectively, which occur alternately midway between the points where the functions is zero.

SINE FUNCTION COSINE FUNCTION

QUADRANT

AS S VARIES
VALUES OF SIN S
VALUES OF COS S

I
0 to pie/2
0 to 1
1 to 0

II
Pie/2 to pie
1 to 0
0 to (-1)

III
Pie to 3pie/2
0 to (-1)
(-1) to 0

IV
3pie/2 to 2pie
(-1) to 0
0 to 1

Noting the quadrant in which terminal point P(S) lies, you are able to determine the algebraic sign of the sine and cosine functions in the different quadrants as indicated in the table below.

QUADRANT

SIN S
COS S
I

+
+
II

+
-
III

-
-
IV

-
+

THIS IMPLIES THAT IN :

QUADRANT I – 0 <S<π/2 0<SIN S<1 0<COS S<1
QUADRANT II – 2<S<π 0<SIN S<1 -1<COS S<0
QUADRANT II – π <S<3π -1<SIN S<0