# Trigonometric Functions-I, Trigonometric Functions, Relation Between Length of an Arc and Radius of the Circle

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## Relation between Length of an Arc and Radius of the Circle

An angle of radian is subtended by an arc whose length is equal to the radius of the circle. An angle of radians will be substened if arc is double the radius.

An angle of radians will be subtended if arc is times the radius.

All this can be read from the following table:

Therefore, or , where radius of the circle, angle substended at the centre in radians, and length of the arc.

The angle subtended by an arc of a circle at the centre of the circle is given by the ratio of the length of the arc and the radius of the circle.

Example:

Find the angle in radians subtended by an arc of length at the centre of a circle of radius .

Solution:

and .

radians or

radians, or

radians

## Trigonometric Functions

While considering, a unit circle you must have noticed that for every real number between and , there exists a ordered pair of numbers and . This ordered pair represents the coordinates of the point .

If we consider on the unit circle, we will have a point whose coordinates are . If

, then the corresponding point on the unit circle will have its coordinates . In the above figures you can easily observe that no matter what the position of the point, corresponding to every real number we have a unique set of coordinates .

The values of and will be negative or positive depending on the quadrant in which we are considering the point. Considering a point (on the unit circle) and the corresponding coordinates , we define trigonometric functions as:

(for ) , (for )

(for ) , (for )

Now let the point moves from its original position in anti-clockwise direction. For various positions of this point in the four quadrants, various real numbers will be generated. We summarise, the above discussion as follows. For values of in the:

I quadrant, both and are positve.

II quadrant, will be negative and will be positive.

III quadrant, as well as will be negative.

IV quadrant, will be positive and will be negative.

or I quadrant II quadrant III quadrant IV quadrant

All positive positive positive positive

positive positive positive

Where what is positive can be rememebred by:

All

Quardrant I II III IV

Example: What will be sign of the following?

(i) (ii) (iii)

Solution:

(i) Since lies in the first quadrant, the sign of will be positive.

(ii) Since lies in the first quadrant, the sign of will be positive.

(iii) Since lies in the first quadrant, the sign of will be negative.