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
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.

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