If cos θ = 2/3 and 270° < θ < 360° then determine the exact value of 1/cot

Q4. cosA = 2/3 = X/r,

X^2 + Y^2 = r^2,
2^2 + Y^2 = 3^2,
4 + Y^2 = 9,
Y^2 = 9 - 4 = 5,
Y = +-sqrt5,
Y = -sqrt5(Q4).

1/cotA = tanA = Y/X = -(sqrt5/2).

To find the value of 1/cot θ, we first need to find the value of cot θ.

Given that cos θ = 2/3, we can use the Pythagorean identity to find sin θ:

sin^2 θ + cos^2 θ = 1
sin^2 θ + (2/3)^2 = 1
sin^2 θ + 4/9 = 1
sin^2 θ = 1 - 4/9
sin^2 θ = 5/9
sin θ = √(5/9)
sin θ = √5/3

Since θ is in the third quadrant (270° < θ < 360°), sin θ is negative in this quadrant, so:

sin θ = -√5/3

Now, we can find cot θ by taking the reciprocal of tan θ:

cot θ = 1/tan θ

Since tan θ = sin θ / cos θ, we can substitute the values we have:

cot θ = 1 / (sin θ / cos θ)

cot θ = cos θ / sin θ
cot θ = (2/3) / (-√5/3)
cot θ = -2√5/5

Finally, we can find 1/cot θ:

1/cot θ = 1 / (-2√5/5)

To divide by a fraction, we can multiply by its reciprocal:

1/cot θ = 1 * (5/-2√5)
1/cot θ = -5/2√5

Therefore, the exact value of 1/cot θ is -5/2√5.

To determine the exact value of 1/cot, we need to first find the value of cot θ.

cot θ is the reciprocal of tan θ, which is equal to the ratio of cosine to sine.

Given that cos θ = 2/3, we can use the Pythagorean identity to find the value of sin θ:

sin θ = √(1 - cos^2 θ)
sin θ = √(1 - (2/3)^2)
sin θ = √(1 - 4/9)
sin θ = √(9/9 - 4/9)
sin θ = √(5/9)
sin θ = √5 / 3

Now, we can find the value of cot θ:

cot θ = cos θ / sin θ
cot θ = (2/3) / (√5 / 3)
cot θ = 2 / (√5)

Finally, to find the value of 1/cot, we take the reciprocal of cot θ:

1/cot θ = 1 / (2 / (√5))
1/cot θ = (√5) / 2

Therefore, the exact value of 1/cot is (√5) / 2.