Expert: Steve Holleran - 4/30/2007

Question
Hi,
I'm confused as to how to find the value of the (5) other trig functions when given the following info:
cos(pheta)= -1/2 and lies in quadrant II.

My second question is, how do I determine whether the given trig. function is positive, negative, zero or undefined?
1. cos [(pi)/4]
2. tan 90 degrees


Thank you in advance!!

Answer
Hi Nik,

I'm going to use "A" instead of "pheta" because its easier for me, ok?

In the first question, since A lies in quadrant II, draw an angle with terminal side in II.  Now drop an altitude to the negative x-axis.  Now you have a right triangle.
Since the cos A = -1/2, and cos = adjacent/hypotenuse,
use the reference angle formed by the terminal side and the negative x-axis as A , and label the side of the triangle along the negative x-axis (the adjacent side)
as -1, and the terminal side of the angle (the hypotenuse) as 2.  Using Pythagorean Theorem, the remaining side, the altitude you drew, should be sqrt(3).

An important point is that the adjacent side is -1, since you are in Quad II, and the opposite is positive 2.

Now just use SOH CAH TOA and read off the other functions, using reference angle A, then get the reciprocal functions:

cos A = -1/2 ----------> sec A = -2

sin A = sqrt(3)/ 2-----> csc A = 2/sqrt(3)

tan A = sqrt(3) / -1---> cot A = -1/sqrt(3)


2.  Here, there's a nice device you can use to help remember the sign of the basic trig functions in each quadrant. I'll try to "draw" it here:
The lines I've tried to draw represent the x and y axes.

                   S  |   A
                -------------
                   T  |  C     

For the sine, cosine and tangent, the diagram tells you in which quadrants each is POSITIVE:

In Q I, they are All positive

In Q II, the Sine is positive

In Q III, the Tangent is positive

In Q IV, the Cosine is positive.

So, for cos(pi/4), since this is in Q I, it it positive.

For the tangent, you have to be very careful at the quadrant positions, because tan = sin / cos, and the sine and cosine functions become 0 at some of them.  Like here,
tan(90) = sin(90) / cos(90), but cos 90 = 0, so tan(90) is undefined.  (It will also be undefined at 270 degrees).


I hope this was of some help.

Steve Holleran