2.3 Functions of General Angles or t Ratios
Draw a unit circle (circle of radius 1 unit).
Let O be the centre of this unit circle. Set a rectangular righthand
coordinate system, taking O as its origin (see fig).
Let the circle cut the xaxis at A º
(1,0). Rotate OA around O (counter clockwise or clock wise) reaching
OP and forming an angle of measure q
.
Obviously OP = 1 unit.
Let P º (x, y)
then we define the following
1. sin q = y / r or
y / 1 = y (\ r = 1)
2. cos q = x / r or
x / 1 = x (\ r = 1)
3. tan q = y / x provided
x ¹ 0
4. cot q = x /y provided
y ¹ 0
5. csc q = r /y = 1/y
provided
y ¹ 0
6. sec q = r/x = 1/x
provided x ¹ 0
Note : Whenever we wish to
find the values of t ratios of standard angles or prove some result,
we shall say "Draw a unit standard circle and let angle q
be in standard position." This will automatically mean that
the initial side of the angle is along the xaxis, vertex being
at the origin.
The Signs Of Trigonometric Ratios
Since r is always positive, the signs of the functions
in various quadrant depend upon the signs of x and y. To determine
these signs one may visualize the angle in standard position.
1) If P(x, y) be in the 1st quadrant both x and
y are positive. Hence all tratios are positive.
2) If P (x, y) be in the IInd quadrant x is negative
and y is positive.
\ sin q
= y / r , csc q = r / y are positive
cos q = x / r, sec
q = r / x, tan q
= y / x and cot q = x / y are all negative
3) If P (x,y) be in the IIIrd quadrant, both x
and y are negative.
Hence sin q y /r , cos q
= x / r, csc q = r / y
and sec q r / x are all negative whereas
tan q = y / x and cot q
= x / y are positive.
4) If P(x, y) be in the IVth quadrant, x is positive
but y is negative.
Hence cos q = x / r
and sec q = r /x are positive but sin
q = y / r, csc q
= r / y, tan q = y /x and cot
q = x/y are all negative.
A crude aid to remember the signs is the four letter
acronym (phrase) ASTC which stands for All Silver Tea Cups or "Arizona State Teacher's College" or any other four worded
phrase that will help you to remember the relationships.
Note:
A 
S 
T 
C 
¯ 
¯ 
¯ 
¯ 
All are +ve In Ist quad.

sine and csc. are +ve in II quad. 
tan and cot are +ve in III quad 
cos and sec are + ve in IV quad 
Trigonometric Ratios Of Quadrantal Angles
For these angles the terminal side of an angle
coincides with one of the coordinate axes. Therefore point P(x,
y) on the terminal side has either x = 0, y ¹
0 or x ¹ 0, y = 0. In either case,
two of the six tratios are not defined. For instance, for angle
00 terminal side coincides with +ve of xaxis and its ordinate i.e
y = 0. Now cot q = x / y and csc q
= r / y, having y = 0 in the denominator are not defined. Some authors
write this as cot 0^{0} = ±
¥ or simply cot 0^{0} = ¥
.
The t ratios of the quadrantal angles are :
Click here to enlarge
