Answer:
cot(θ) = 1
Explanation:
It is helpful to have knowledge of the definitions of the trig functions. The mnemonic SOH CAH TOA can help, but you also need to remember the inverse functions, csc, sec, cot.
(I remember that the function and its inverse differ in their co- prefix. COsecant is the inverse of sine, which has no CO-, while COsine is the inverse of secant, which has no CO-. That's my memory crutch. Yours may be different.)
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When given a ratio and a quadrant, as in this problem, it can be helpful to draw the associated triangle. See the attachment.
The cosecant is the inverse of the sine function so its ratio is ...
csc = hypotenuse/opposite
The given value for csc is -√2, so we can label the hypotenuse √2 (it is convenient for the hypotenuse to always be positive). The opposite side can be labeled -1 so the ratio comes out right. Since this is a 3rd-quadrant angle, the adjacent side also has length -1. (You can figure it out using the Pythagorean theorem if you don't already recognize the triangle as being isosceles.)
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We have marked the triangle to correspond to csc(θ) = -√2. Now, we need to find cot(θ).
The mnemonic tells us the ratio for tangent, and we know the cotangent is its inverse:
cot = adjacent/opposite
cot(θ) = -1/-1
cot(θ) = 1
This is consistent with our knowledge of the sign of the tangent function in the third quadrant (positive).
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It can work reasonably well to fill in dimensions of the triangle consistent with the given trig ratio, then use the dimensions to find the required ratio. Often, you will have to use the Pythagorean theorem to find the third side of the triangle. It can be useful to remember that ...
- sin > 0 for y > 0
- cos > 0 for x > 0
The signs of the other functions can be derived from these.