If cos⁡ θ =-1/2 and tan⁡ θ > 0, then find the exact value of csc⁡θ.

To find the exact value of csc θ, we first need to understand what the given conditions tell us about θ.

1. cos θ = -1/2: This tells us that θ is in the second or third quadrant (where cosine is negative).

2. tan θ > 0: This tells us that θ is in the first or third quadrant (where tangent is positive).

Combining these two pieces of information, we can conclude that θ is in the third quadrant (where both cosine is negative and tangent is positive).

Now, we know that cos θ = adjacent/hypotenuse. Given that cos θ = -1/2, we can consider a right triangle where the adjacent side is -1 and the hypotenuse is 2.

Using the Pythagorean theorem, we can find the opposite side: sqrt(2^2 - (-1)^2) = sqrt(3).

Since θ is in the third quadrant, the opposite side is also negative, so we have -sqrt(3).

The cosecant function is defined as csc θ = hypotenuse/opposite. Substituting the values we found, we get csc θ = 2 / -sqrt(3).

To rationalize the denominator, we multiply the numerator and the denominator by sqrt(3) to get csc θ = -2sqrt(3) / 3.

So, the exact value of csc θ is -2sqrt(3) / 3.