Find sin θ if cot θ = – 4 and cos θ < 0.?

Answer 1

cos θ < 0 and cot θ < 0 means that θ must be more than 90° and less than 180° in other words must be sin θ > 0

if cot θ = – 4 then (cot θ)² = 16
(tan θ)² = 1/16
(sin θ)²/(cos θ)² = 1/16
(sin θ)² = (1/16)(cos θ)²
(sin θ)² = (1/16)(1 – (sin θ)²)
multiply both sides by 16
16(sin θ)² = 1 – (sin θ)²
17(sin θ)² = 1
(sin θ)² = 1/17
sin θ = 1/√17

Answer 2

Since cot is negative, where is cot negative? Second and fourth quadrants.

Now we need to fins where cos is negative. That’s easy, the second quadrant is what we are looking for. This tells us that sin is positive.

Now, if cot x = -4, that must mean that tan x = -1/4. That would be opposite over adjacent.

Since sin x requires the hypotenuse, find the hypotenuse:

z^2 = 1^2 + 4^2

z^2 = 1 + 16

z = sqrt 17

Since sin is positive, we end up with:

1/(sqrt 17), or (sqrt 17) / 17

Miss Kristin

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