Find cotangent theta given that cosecant theta equals -3.5891420 and theta is in the third quadrant.

I was using the trig identity 1+cot^2theta=csc^2theta

I wanted to isolate cotangent so I plugged in 1/sin (-3.5891420) and then squared my answer. I then subtracted one from both sides and sqaured both sides. I am getting 15.9427 when the answer is 3.44701905

please let me know where I am going wrong.

since 1+cot^2theta=csc^2theta

cot^2 Ø = csc^2 - 1
= (-3.589142)^2 - 1
= 11.8819403
cotØ = ± √11.8819403
= ± 3.447...
but since Ø in in III, cotØ must be positive, so
cotØ = 3.447..

Why were you doing
1/sin(-3.58914...) ????

-3.58... is NOT an angle, the csc(of some angle ) = -3.58...

To find cotangent theta, we can use the identity cot^2theta = csc^2theta - 1. However, it seems there might be a mistake in your calculation.

Let's start over and follow the steps correctly:

Given: csc(theta) = -3.5891420

Step 1: Determine the sine(theta) from the given cosecant(theta) value.
Recall that csc(theta) = 1/sin(theta), so sin(theta) = 1/csc(theta).
Therefore, sin(theta) = 1/-3.5891420 = -0.2781 (rounded to four decimal places).

Step 2: Determine the cosine(theta) using the Pythagorean identity.
Since theta is in the third quadrant, both sine and cosine will be negative. We can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 to find the cosine.
Square the value of sine we obtained earlier: (-0.2781)^2 = 0.0773
Now, cos^2(theta) = 1 - sin^2(theta) = 1 - 0.0773 = 0.9227.
Therefore, cos(theta) = -√0.9227 (note the negative sign because theta is in the third quadrant) ≈ -0.9603 (rounded to four decimal places).

Step 3: Use the definition of cotangent(theta) to find the cot(theta).
cot(theta) = cos(theta) / sin(theta)
Substituting the values, we get cot(theta) = (-0.9603) / (-0.2781) ≈ 3.4470 (rounded to four decimal places).

So the correct value for cotangent theta, given that cosecant theta is -3.5891420 and theta is in the third quadrant, is approximately 3.4470.