Determine the values of sin, cos, tan, csc, sec and cot at (-3, -3) on the terminal arm of an angle in standard position

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

1. tanAr = Y/X = =-3 / -3 = 1.00,

Ar = 45deg. = Ref. angle.

A is in 3rd Quadrant.
A = 180 + 45 = 225deg.

sin(225)= -0.7071,

cos(225) = -0.7071,

tan(225) = 1.00,

csc(225) = 1/sin(225) =

sec(225) =

cot(225)

You can finish it.

2. Theta is bet. 270 and 360deg.

Given: cos(theta) = 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 = sqrt(5).

1/cot(theta) = tan(theta),
1/cot(theta) = Y/X = sqrt(5) / 2.

To determine the trigonometric values at (-3, -3) on the terminal arm of an angle in standard position, we can use the given coordinates to find the distance from the origin (r) and the signs of x and y (positive or negative) in the respective quadrants.

1. Finding the values at (-3, -3):
First, find the hypotenuse (r) using the Pythagorean theorem:
r = √((-3)^2 + (-3)^2) = √(9 + 9) = √18 = 3√2

Now, we can determine the trigonometric values:
sin θ = y/r = -3/(3√2) = -√2/2
cos θ = x/r = -3/(3√2) = -√2/2
tan θ = y/x = -3/-3 = 1
csc θ = 1/sin θ = -2/√2 = -√2
sec θ = 1/cos θ = -2/√2 = -√2
cot θ = 1/tan θ = 1/1 = 1

Therefore, at (-3, -3) on the terminal arm of an angle in standard position, the trigonometric values are:
sin θ = -√2/2
cos θ = -√2/2
tan θ = 1
csc θ = -√2
sec θ = -√2
cot θ = 1

2. Given cos θ = 2/3 and 270° < θ < 360°, we need to determine the exact value of 1/cot θ.
To find cot θ, we first need to find tan θ using the given values.
tan θ = sin θ / cos θ = (√(1 - cos^2 θ)) / cos θ
Since θ is in the fourth quadrant (270° < θ < 360°), sin θ will be negative.

Let's perform the calculations:
sin θ = -√(1 - (2/3)^2) = -√(1 - 4/9) = -√(5/9) = -√5/3

Now we can find tan θ:
tan θ = sin θ / cos θ = (-√5/3) / (2/3) = -√5/2

Finally, to find 1/cot θ:
1/cot θ = 1 / (1/tan θ) = tan θ = -√5/2

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