find sin of theta if cos is less than 0 and cot equals 3

Answer 1

cosθ < 0 means θ is in 2nd or 3rd quadrant
cotθ > 0 means θ in 1st or 3rd quadrant.

So, θ is in 3rd quadrant

cot 18.4° = 3

so, θ = 180 + 18.4 = 198.4°

sin 198.4° = 0.316

or, more directly, if cotθ = 3, sinθ = -1/√10 = 0.316

Answer 2

Oops. Both those values should be -0.316

Answer 3

cos ( theta ) = + OR - sqrt [ 1 - sin ^ 2 ( theta ) ]

cot( theta ) = cos ( theta ) / sin ( theta )

if cos ( theta ) < 0 that mean cos ( theta ) is negative

if cot ( theta ) = 3 and cos ( theta ) is negative sin ( theta ) must be negative.

So:

cos ( theta ) / sin ( theta ) = 3

sqrt [ ( 1 - sin ^ 2 ( theta ) ] / sin ( theta ) = 3

[ ( 1 - sin ^ 2 ( theta ) ] / sin ^ 2 ( theta ) = 9

1 - sin ^ 2 ( theta ) = 9 sin ^ 2 ( theta )

1 = 9 sin ^ 2 ( theta ) + sin ^ 2 ( theta )

1 = 10 sin ^ 2 ( theta )

10 sin ^ 2 ( theta ) = 1 Divide both sides with 10

sin ^ 2 ( theta ) = 1 / 10

sin ( theta ) = + OR - sqrt ( 1 / 10 )

If sin ( theta ) is negative solution is :

sin ( theta ) = - 1 / sqrt ( 10 )