Circular Functions

Theorem 4

Statement:

If 'n' be any integer then,

i) sin (2pn + q) = sin q

ii) cos (2pn + q) = cos q

Proof:

Draw a unit circle with centre 'O'.

Let XOX' and YOY' be the coordinate axes.

Let a variable point A on the circumference of the circle move from the position A (1,0) to the final position P(q) along the circumference of the circle.

Let AP be the arc, so that arc AP subtends angle q at the centre. Let us suppose that the variable point A makes n complete rotations and further moves through arc length q. Then finally it comes to the position P.

Then arc length covered by the variable point along the circumference is (2np + q).

= abscissa of the P(q)

=

= ordinate of the P(q)

= sin q

Note: