Trigonometry XI


   
 
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:
 
 
 
 
 
 
 
 
 
 
 
 
 
  
     
   
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