Trigonometry XI


   
 
Trigonometric Functions
Unit Circle
 
The circle whose radius is 1 unit whose centre is the origin of a rectangular co-ordinate system is called the unit circle.
 
 
 
 
Let q be any real number. Begin at A(1, 0) on the unit circle and measure along the circumference of arc length |l| units. If q > 0 measure the arc in the anti-clockwise direction. If q < 0, measure the arc in the clockwise directions.
 
This locates a unique point on the circumference of the unit circle. We call the point then locates as the trigonometric point P(q). The real number q and the point P(q) forms an ordered pair. This process consequently defines a function whose domain as the set of all real numbers and whose range is the set of all points on the unit circle. We write the ordered pairs of this function as (q, P(q)).
 
Let q be any real number and let the co-ordinates of P(q) be (x,y). We define the cosine function of q (written as cosq) and sine function of q (written as sin q).
 
In term of the co-ordinates of the trigonometric point.
 
Definition
 
If the trigonometric point P(t) has co-ordinates (x, y) then cos q = x, sin q = y.
 
 
The point P (x,y) on the unit circle can be written as P (cos q, sin q).
 
From the figure,
 
OP2 = OM2 + PM2
 
x2 + y2 = 1
 
Now we have the identity (cos q)2 + (sin q)2 = 1.
 
 
 
 
Further four additional functions are defined as follows:
 
 
 
 
 
 
Now the above six function can be expressed in terms of x and y.
 
 
 
 
 
 
 
The six functions of q defined by the above equation are called trigonometric functions of q or circular functions of q.
 
 
Now consider the unit circle of radius 1 unit. Then the circumference is 2p. If a moving point A starts from A and travel in the counter clockwise direction then at the points A(1, 0), B(0, 1), A'(-1, 0) and B'(0, -1) and A(1, 0), the angles of rotation or the arc length covered are respectively.
 
 
 
 
 
 
 
 
 
 
Consider the unit circle as shown below:
 
 
 
Trisect the arc AB at the points P and Q, so that
 
Arc AP = Arc PQ = Arc QB
 
 
(Here arc AB is as same as angle subtented by arc AB at the centre)
 
 
Draw PM and QN perpendicular to x-axis.
 
In D OMP, we have
 
 
From geometry in D MOP,
 
 
 
 
 
Again in DQON,
 
OQ = 1 = radius of unit circle,
 
 
 
 
 
 
 
 
 
 
Draw PM perpendicular to x-axis.
 
 
Let OM = l = PM.
 
From the right angled DOMP OM2 + PM2 = 1
 
l2 + l2 = 1
 
2l2 = 1
 
 
 
 
 
Theorem 3
 
Statement:
 
sin2q + cos2q = 1
 
Proof:
 
 
Let X'OX and Y'OY be the rectangular co-ordinate axes with O as centre and radius unity draw a circle cutting OX at A and OY at B.
 
Let the moving point A move to position P so that arc AP = q. Let the coordinate of P be (x, y). By the definition x2 + y2 = 1 and x = cos q,
 
y = sin q.
 
\ sin2q + cos2q = 1
 
Some Important Results
 
a) Prove cot2q + 1 = cosec2q.
 
We know that cos2q + sin2q = 1 ...(1) also
 
 
Divide (1) by sin2q on both sides (sinq 0), then
 
 
 
b) Prove that 1 + tan2q = sec2q.
 
 
 
 
 
The following has to be committed to memory
 
1) sin2q + cos2q = 1
 
 
 
2) sec2q - tan2q =1,
 
 
 
3) cosec2q - cot2q = 1
 
 
 
 
 
 
 
  
     
   
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