Question

For what values of theta belongs to (0,2) is the expression 1/ 1 + sec theta undefined?

a) 2 pi/3, 4 pi/3
b) pi/2, 3pi/2
c) 0
d) 5 pi/6 , 7pi/6

Answer 1

Hey, Tomcat, I suspect your interval is (0,2pi), and not (0,2). Double-check, would you please?

Answer 2

It says (0, 2) but it might be wrong

Answer 3

Here's the original expression:\[\frac{ 1 }{ 1+\sec \theta }\]

Answer 4

This expression will be undefined for any theta for which the denominator, 1+sec theta, is zero, right?
So, we must solve \[1+\sec \theta=0\]

Answer 5

Sorry, I made a mistake, it supposed to be 1 / 2 + sec theta

Answer 6

How would you go about solving this equation for theta, restricting theta to the interval (0,2pi)?

Answer 7

OK: Then we must now solve \[2+\sec \theta=0\]

Answer 8

How?

Answer 9

sec theta = - 2, but it will not be in interval 0,2 pi

Answer 10

TomKat: Here's what I'd do:\[2+\sec \theta=0\]
\[\sec \theta=\frac{ -2 }{ 1 }\]
\[\sec \theta=\frac{ 1 }{ \cos \theta }\]

Answer 11

\[\frac{ -2 }{ 1 }=\sec \theta=\frac{ 1 }{ \cos \theta }\]

Answer 12

Can you now solve this for theta? How many solutions would you expect, and why?

Answer 13

Equivalently, \[\cos \theta=-1/2\]

Answer 14

Solutions?

Answer 15

theta = - 1/2 / cos

Answer 16

TomKat: the cosine function is an operator (a command); it can't be separated from the angle (theta) in this manner.

Review the definitions of the 3 basic trig functions. In particular, note that \[\cos \theta=\frac{ adjacent side }{ hypotenuse }=\frac{ adj }{ hyp}\]

Answer 17

Does that look at all familiar to you?

Answer 18

yes

Answer 19

Good.
Then perhaps it'd make more sense to write something like this:
\[\cos \theta=\frac{ adj }{ hyp }=\frac{ -1 }{ 2 }\]
Could you draw one or more triangles within a circle for which this relationship is true? In other words, choose the appropriate quadrants (there are 2 that apply here), and draw in each quadrant a triangle with adjacent side -1 and hypotenuse 2?

Answer 20

If you're able to do that, it shouldn't be hard to identify the angles involved. We're trying to solve for the values of those two angles.

Answer 21

that will apply to quadrant 2 and 3, thats where cos is negative