θ equals the angle in Quadrant II so that cot θ= -3. Find the values for sin θ, cos θ, and tan θ.

Draw your triangle

x = -3
y = 1
r = √10

Now use the definitions of the trig functions to fill in the table.

To find the values for sin θ, cos θ, and tan θ, we can use the fact that cot θ is the reciprocal of tan θ. Since cot θ is given as -3, we can find tan θ by taking the reciprocal, which gives us tan θ = -1/3.

Since θ is in Quadrant II, the sine of θ will be positive, while the cosine will be negative. To find sin θ and cos θ, we will use the Pythagorean identity: sin^2θ + cos^2θ = 1.

Let's start with finding cos θ. We can rewrite the equation as:

1 - sin^2θ = cos^2θ

Since we know that cos θ is negative, we will use the negative square root:

cos θ = -√(1 - sin^2θ)

Next, we substitute the value of tan θ into the equation:

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

Simplifying further:

cos θ = -√(8/9) = -2√(2) / 3

Now, we can find sin θ. We'll use the Pythagorean identity again, but this time solve for sin θ:

sin^2θ = 1 - cos^2θ
sin^2θ = 1 - (-2√(2)/3)^2 = 1 - 8/9

Simplifying:

sin θ = √(1 - 8/9) = √(1/9) = 1/3

Finally, we can determine the values for sin θ, cos θ, and tan θ:

sin θ = 1/3
cos θ = -2√(2)/3
tan θ = -1/3