I have a question I have been working on since yesterday and I am not making this up. I couldn't get the right answer.

If sin theta = -2/3, which of the following are possible?
A: cos theta= -the sqr rt of 5/3 and tan theta =2/3.

B: sec theta = 3/the sqr rt of 5 and tan teta = -2/the sqr rt of 5.

c: sec theta =-3/2 and tan theta =2/the sqr rt of 5.

D: cos theta= the sqr rt of 5/3 and tan theta = 2/the sqr rt of 5.

My work:
Sin theta= x/r
cos theta = y/r
tan theta = y/x
sec theta = r/y

Pythagorean theorem:
X^2 +y^2 = r^2
y= -2, r= -3
x= The sqr rt of (-3^2 -(-2^2))=
Sqr rt of (9-4)= Sqr rt of 5
x= sqr rt of 5

sin theta = -sqr rt of 5/3
cos theta = -2/3
tan theta = -2/ the sqr rt of 5
sec theta = -3/2

ok I tried putting my work into the answers yet I couldn't get it right. What am I doing wrong? Pls help.

sinθ = y/r

So, y is negative, meaning QIII or QIV

Just working in QI, we see that if sinθ = 2/3,
cosθ = √5/3
tanθ = 2/√5

Now check in QIII and QIV to see what's possible. Looks like (B) in QIV to me.

Look closely at the other answers and verify that they fail to satisfy the conditions.

I got it wrong. I don't understand why.

r is never negative -- it is just the distance from (0,0).

Plus, they said right at first, that sinθ = -2/3, so how did you come up with sinθ = -√5/3?

Draw two triangles, one in QIII and one in QIV where y = -2 and r = 3.

Then you will see why only (B) can be.

To determine which of the given options is possible, we need to evaluate whether the values of sin(theta), cos(theta), tan(theta), and sec(theta) match the given values.

Given:
sin(theta) = -2/3

First, we can find cos(theta) using the Pythagorean identity:
cos(theta) = sqrt(1 - sin²(theta))
cos(theta) = sqrt(1 - (-2/3)²)
cos(theta) = sqrt(1 - 4/9)
cos(theta) = sqrt(5/9)
cos(theta) = sqrt(5)/3

Next, let's calculate tan(theta):
tan(theta) = sin(theta)/cos(theta)
tan(theta) = (-2/3) / (sqrt(5)/3)
tan(theta) = -2/sqrt(5)

Now, let's check each option:

Option A:
cos(theta) = -sqrt(5/3) [Incorrect, should be positive]
tan(theta) = 2/3 [Incorrect, should be negative]
So, Option A is not possible.

Option B:
sec(theta) = 3/sqrt(5) [Correct]
tan(theta) = -2/sqrt(5) [Correct]
So, Option B is a possible solution.

Option C:
sec(theta) = -3/2 [Incorrect, should be positive]
tan(theta) = 2/sqrt(5) [Correct]
So, Option C is not possible.

Option D:
cos(theta) = sqrt(5)/3 [Correct]
tan(theta) = 2/sqrt(5) [Correct]
So, Option D is a possible solution.

Based on our calculations, the possible options are B and D.