Question

Please Help Me!! Will Fan!
So i kind of understand how to find the trigonometric functions but i really need help on writing the domain
(sin theta) (sec theta)= (tan theta)

Answer 1

the problem is labeled as number 8, i dont know how to write the domain. this is not my work

Answer 2

what is sec(theta) in terms of sine or cosine?

Answer 3

1/cos theta

Answer 4

because we have a fraction with a variable term in the denominator, this means there is a possibility that the denominator could be 0

Answer 5

when is the denominator zero?

Answer 6

never!

Answer 7

or else it'll be undefined, right?

Answer 8

i really dont understand how to find the domain of validity

Answer 9

you have the right idea

the denominator *should* not be zero to make sure you're not dividing by zero (since it's undefined)

but there is a possibility of a variable expression being equal to zero

Answer 10

for example

1/(x-6)

is undefined when x = 6 because this value makes the denominator zero

Answer 11

oh i see

Answer 12

if we restrict that x cannot equal 6, then we avoid this division by zero error

Answer 13

similarly, cos(theta) could be equal to zero at certain values of theta

Answer 14

when is cos(theta) = 0 true?

Answer 15

im not too sure, does it equal zero at 1?

Answer 16

do you have a unit circle?

Answer 17

oh, is it pi/2?

Answer 18

because that's the first radian on the cosine graph?

Answer 19

good, cos(pi/2) = 0

Answer 20

what other values do you find?

Answer 21

pi, 3pi/2, and 2pi

Answer 22

3pi/2, yes

2pi, no

Answer 23

cos(2pi) = 1

Answer 24

5pi/2?

Answer 25

so what's a general rule to capture all of the solutions of cos(theta) = 0

Answer 26

all odd multiples of pi/ 2

Answer 27

that's one way, what's another?

Answer 28

all reals except odd multiples of pi/2?

Answer 29

well another way is to start with pi/2

then add on multiples of pi (so pi/2 + pi = 3pi/2 or pi/2 + 2pi = 5pi/2, etc etc)

Answer 30

so one complete solution set is

\[\Large \theta = \frac{\pi}{2} + \pi*n\]

where n is any integer

Answer 31

ohh, okay that makes a lot of sense

Answer 32

going back to the original equation, theta can be any value you want but it cannot be values in the form of \[\Large \theta = \frac{\pi}{2} + \pi*n\] where n is any integer

Answer 33

this is to make sure you don't divide by zero