1 Feb 1999

rigonometric Ratios

The diagram shows a circle of radius r with its centre at the origin.

A rotating radius OP rotates through an angle q from the x-axis.

The angle q is
  1. positive if it is rotated in the anticlockwise direction,
  2. negative if it is rotated in the clockwise direction.
Let (x, y) be the coordinates of the point P.� The trigonometric functions are defined as follows.

sin q =�
y
�
r
����� cosec q =�
r
�
y
cos q =�
x
�
r
sec q =�
r
�
x
tan q =�
y
�
x
cot q =�
x
�
y

pecial Angles

x
�0�
p/6
p/4
p/3
p/2
� p�
3p/2
�2p�
sin x
0
1
�
2
1
�
�2
�3
�
2
1
0
-1
0
cos x
1
�3
�
2
1
�
�2
1
�
2
0
-1
0
1
tan x
0
1
�
�3
1
�3
--
0
--
0

asic Angles

The basic angle is the acute angle between a rotating radius and the x-axis.

Thus 0� � basic angle � 90�.

igns Of Trigonometric Ratios

If a is the basic angle of q, then

omplementary Angles

If the sum of two acute angle is 90�, they are said to be complementary angles of each other.

In general, for any angle q,

sin (90� - q)
 �= cos q
cos (90� - q)
 �= sin q
sec (90� - q)
 �= cosec q
����� cosec (90� - q)
 �= sec q
tan (90� - q)
 �= cot q
cot (90� - q)
 �= tan q

raphs Of Trigonometric Functions

The graphs of sin x, cos x and tan x should be memorised.� Other trigonometric graphs may be generated from them by translations, scalings, reciprocal or a mixture of these transformations.

  1. y = sin x
    2.

    1. The graph is continuous, ie, it has no breaks.
    2. The range is -1 � y � 1.
    3. It is periodic with a period of 2p, ie, sin (x + 2p) = sin x.

  2. y = cos x

    1. The graph is continuous.
    2. The range is -1 � y � 1.
    3. It is periodic with a period of 2p.
    4. It can be obtained from the graph of y = sin x by a translation to the left by p/2, ie, cos x = sin (x + p/2).

  3. y = tan x

    1. It is not continuous, being undefined at x = (2n + 1)p/2 for all n � Z.
    2. The range is R.
    3. It is periodic with a period of p.

To Other Sites

Hosted by www.Geocities.ws

1