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
¾
r
      cosec q = 
r
¾
y
cos q = 
x
¾
r
sec q = 
r
¾
x
tan q = 
y
¾
x
cot q = 
x
¾
y

pecial Angles

x
 0 
p/6
p/4
p/3
p/2
  p 
3p/2
 2p 
sin x
0
1
¾
2
1
¾
Ö2
Ö3
¾
2
1
0
-1
0
cos x
1
Ö3
¾
2
1
¾
Ö2
1
¾
2
0
-1
0
1
tan x
0
1
¾
Ö3
1
Ö3
--
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 Î Z.
    2. The range is R.
    3. It is periodic with a period of p.

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