Even and Odd functions

i) A function f(x) is said to be even if f(-x) = f(x)

ii) A function f(x) is said to be odd if f(-x) = -f(x)

e.g., f(x) = x³, f(-x) = (-x)³ = -x³ = -f(x)

f(x) = cos x is even for f(-x) = cos (-x) = cos q = f(x)

f(x) = x cos x is odd for f(-x) = (-x) cos (-x) = -x cos x = -f(x)

f(x) = sin x is odd whereas f(x) = x sin x is even.

Theorem 6

Statement:

For all real numbers x and y

Let x and y be any two real numbers and let P(x)and Q(y) be the corresponding trigonometric points on the unit circle. In the above figures we have taken x and y so that

The coordinates of P(x) are (cosx, sin x) and that of Q(y) are (cos y, sin y) by the definitions of cosine and sine functions.

Now choose a point R on the unit circle so that arc AR has a measure of (x-y) units. Then the trigonometric point with respect to (x-y) is R(x-y) and the corresponding coordinates of R are (cos(x-y),sin(x-y)). The arc length of AR is the same as arc PQ and hence the chord lengths of PQ and AR are same.

i) |AR| = distance from R to A.

Since AR = PQ, we have

Theorem 7

Statement:

For all values of

Proof:

Theorem 8

For all real values of x.

Proof:

Theorem 9

For real values of x

Proof:

Theorem 10

Proof:

Theorem 11

Proof:

Now put x = y, then

Now put x = y, then

=

Theorem 12

Proof:

Adding (1) and (2)

Subtracting (2) from (1)

Adding (3) and (4)

Subtracting (4) from (3)

Theorem 13

Proof:

Theorem 14

Proof:

For all A and B we have

Let A + B = x, A - B = y, then

Adding (b) and (a), we get

Subtracting (b) from (a), we get

Adding (c) and (d), we get

Subtracting (d) from (c), we get

Theorem 15

Proof: