Question

Can someone help me figure out this? What is the domain of validity of the trigonometric identity csc theta = 1 / sin theta?

Answer 1

@UnkleRhaukus @MathLegend

Answer 2

@Hero

Answer 3

I know it is a reciprocal identity I just cannot figure out how to find the domain of validity

Answer 4

Well, we know sin theta cannot equal zero. So find the angles that would make sin theta equal zero.

Answer 5

when I type it into my calculator, when x or theta =0 I get an error ?

Answer 6

What did you type into your calculator?

Answer 7

I typed 1/sin (theta)

Answer 8

into a graphing calculator, I looked at the table and it said error for x being 0

Answer 9

Okay, so you know that zero is one of the angles that will not work.

Answer 10

Thats what my graph looks like, it doesn't touch zero at all ?

Answer 11

Do this: Try graphing 1/sin(theta) in one color. Then graph csc(theta) in another color. See if you can notice a distinction between the two graphs.

Answer 12

My calculator doesn't have a csc button,

Answer 13

What kind of calculator do you have?

Answer 14

I'm pretty sure that if you're using a graphing calc, it has one.

Answer 15

TI-84 graphing callculator.

Answer 16

I don't see it anywhere.

Answer 17

Yup, it has one.

Answer 18

Where?! Our teachers been telling us to use the 1/ equation.

Answer 19

Yes, because there is no distinction between csc(theta) and 1/sin(theta)

Answer 20

I understand that.

Answer 21

By the way, I suggest you get a graphing calc with a csc function on it.

Answer 22

They don't give us one for my online school.

Answer 23

Do you go to connections academy?

Answer 24

yes

Answer 25

I see. Seems like everyone does.

Answer 26

I know!!!

Answer 27

@phi

Answer 28

@kymber

Answer 29

What is the domain of validity of the trigonometric identity csc(x)

The domain are all values of x that give a defined value for csc(x) (1/sin(x))
we run into trouble if we take the square root of a negative number (not an issue here) or divide by zero. Dividing by 0 is a problem here: any time sin(x) is 0, 1/sin(x) is a divide by zero. so we must exclude x=0º because sin(0)=0 and 1/sin(0) is not defined.

Answer 30

However, sin(x) is 0 a more than just sin(0), At every 180 degrees (or pi radians) sin is 0

Answer 31

so I've narrowed it down to either
a.the set of real numbers except for zero and odd multiples of pi/2
or
b.the set of real numbers except for zero and odd multiples of pi

Answer 32

am I on the right track?

Answer 33

In radians, we would exclude,
... -3π, -2π, -π, 0, π, 2π, 3π ...

Answer 34

so it would be b?

Answer 35

you should know that pi/2 is ok, because sin(pi/2) = 1 so you are not dividing by 0

Answer 36

Isn't there an option for all multiples of pi?

Answer 37

unless they are being tricky: an even multiple of pi/2 is 2*pi/2 or 4*pi/2 which simplifies to pi, 2pi, etc

Answer 38

I tried telling her to find all the angles that would make sin theta equal zero. I just didn't want to give her the answer.

Answer 39

It is a fine line between helping just enough and too much...

Answer 40

I just thought that 0 would not be included in the domain? along with all the multiples of pi?

Answer 41

How come you didn't say that earlier when I was helping you?

Answer 42

yes, but what choice is that? the 2 you listed don't match that

Answer 43

the set of real numbers
the set of real numbers except for multiples of pi
the set of real numbers except for zero and odd multiples of pi
the set of real numbers except for zero and odd multiples of pi/2

Answer 44

They are being just a little tricky. we can assume that multiples of pi include 0
(we have to, none of the other choices are correct)

odd multiples of pi pi,3pi,5pi... leave out 2pi,4pi, ... which would cause a divide by 0

Answer 45

so it would be the 2nd choice.?

Answer 46

Yes, do you see why?

Answer 47

zero is a multiple of pi because 0*pi = 0

Answer 48

Because even the even multiples cause an invalidity in the equation, not just the odd multiples of pi.

Answer 49

and what causes the invalidity?

Answer 50

thats what I don't understand about this entire unit, I've asked my teacher multiple times and she keeps telling me to reread the section.

Answer 51

It is not too esoteric: dividing by a 0 is not allowed (people don't know what it means)
so the invalid expression would happen if you divide by 0.

Answer 52

Cause of invalidity:

\[\csc(\pi) = \frac{1}{\sin(180^{\circ})} = \frac{1}{0}\]

Answer 53

So, that mean that any number that able to be divided by 0?

Answer 54

if we defined a function f(x)= 1/x this function will not be defined at x=0 (divide by 0 not allowed). Its domain would exclude x=0

same idea for 1/sin(x) except sin(x) is zero at lots of places.. k*pi , k an integer

Answer 55

oh! That diagram actually helps a little.

Answer 56

pi radians = 180 degrees
\[\csc(\pi) = \frac{1}{\sin(\pi)} =\frac{1}{\sin(180^{\circ})} = \frac{1}{0}\]

That should make it a little more clearer

Answer 57

So, that mean that any number that able to be divided by 0?

can you ask again?

Answer 58

so If pi makes the expression invalid, then ANY multiple of pi will also, including zero.

Answer 59

Okay, so no matter what which expression of pi, odd or even, would make the expression invalid?

Answer 60

We know pi is odd
zero degrees equals 2 pi which is even.

Answer 61

Okay, so no matter what which expression of pi, odd or even, would make the expression invalid?

to be clear, I would say makes sin(x) 0 and 1/sin(x) would be 1/0 which is invalid

Answer 62

Thank you, my teacher has not taken any time to help me out and it is really starting to irritate me. I'm two weeks behind in this class because I don't understand it and everytime I ask for clarification she tells me to reread the chapter.