Expert: Josh - 7/20/2004

Question
Name the quadrant in which angle A must lie if secA>0 and cscA<0. My book explains the answer and says,"if secA>0 then cosA>0.If cscA<0 then sinA<0."My question is if sec A>0, logically cosA is not greater than 0.Lets use examples,A=5, so sec(5)=1.0038..., but cos is not equal to that, its equal to .996194...Same goes for csc and sin, and if it was a larger number, there would be even a greater difference in the answer.Please explain their answer as it's probably very ambiguous. Also, if the answer to my own question is that this only applies to the signs in the quadrants of the trigonometric functions and it's a rule, please tell me the rule for tan and cot.And also, please tell me if any of the 3 functions matter if it's greater or less than.Will the rule still apply either way? Like secA<0 is cosa<0, or no?

Answer
Ab,

As I've explained yesterday, there are only two fundamental definitions that you need to remember for trigonometry - how sine(w) and cosine(w) are defined.

Knowing this, tan(w) is simply sin(w)/cos(w).

If you refer to yesterday's diagram,
sine is the ratio between the lengths of the opposite_side (PR) and the hypotenuse (OP).
cos  is the ratio between the lengths of the adjacent side of the triangle (OR) and the hypotenuse (OP).
So, given that diagram, you could write sin(x)=|PR|/|OP|, cos(x)=|OR|/|OP|.

Secant(x) and Cosecant(x) are simply the inverse of the cosine and sine function respectively.
i.e., sec(x) =1/cos(x) = |OP|/|OR| and csc(x)=1/sin(x)=|OP|/|PR|, with respect to that diagram.

Since the periodic functions sin(x) and cos(x) are always bounded between 0 and 1, cosec(x) and sec(x) must be somewhere between 1 and infinity (from the reciprocal of 0). Best to draw diagrams to see this.

Important point: The sign doesn't change just because you take the inverse ratio. For instance, when x=pi/4 rad (or 45 deg), sin(x)=1/sqrt(2). Therefore cosec(x)=1/sin(x)=sqrt(2). They must be simultaneously positive or negative, away from the asymptotes [x=pi/2, 3*(pi/2) etc]
Consider another example. When x=pi, cos(pi)=-1, so, sec(pi)=1/cos(pi)=-1. They are both negative. See the point?

Summary: sec(A)>0 when cos(A)>0, which is in the first and fourth quadrant [0<=A<pi/2 and (3/2)*pi<=A<2*pi respectively]. cosec(A)<0 when sin(A)<0 (ie., in the third and fourth quadrant).

Mneumonic: A.S.T.C. (All stations to central)

meaning in quadrant 1, the interval (0,pi/2], all trig. entities (sin,cos,tan) are positive.
In quad 2, the interval (pi/2,pi], only sine() is positive.
In quad 3, the interval (pi,3pi/2], only tan() is positive.
In quad 4, the interval ((3/2)*pi,2*pi], only cos() is positive.

This effectively answers all your questions,
When is {sin(x) or cos(x) or tan(x)} positive/negative?
Remember "All Stations To Central"