Given that tan θ = - (√3/8) and θ is in QII, find the exact value of

a. r =
b. csc θ
c. cos θ
d. cot θ

a. If you are only given the value of Į , you cannot deduce the value of the radius r.

b. Į = pi - 0.213 rad
= 2.929 rad
csc Į = 4.739

c. cos Į = -0.977

d. cot ƒÆ = 1/(tan ƒÆ )= - (8/(�ã3)

To find the exact values of the trigonometric functions, we can use the given value of tangent (tan θ) and the information about the quadrant (QII) in which θ lies.

a. To find the value of r, we can use the Pythagorean identity for trigonometric functions. In QII, the reference angle (θ') for θ is π - θ. Thus, we have:

sin θ = -√(1 - cos²θ) ...(1)

Given tan θ = - (√3/8), we can express tan θ in terms of sine and cosine:

tan θ = sin θ / cos θ ...(2)

Substituting the values, we have:

(-√3/8) = sin θ / cos θ

Multiplying both sides by cos θ, we get:

-√3 = sin θ ...(3)

Now, we can substitute the value of sin θ from equation (3) into equation (1):

(-√3) = -√(1 - cos²θ)

Squaring both sides, we obtain:

3 = 1 - cos²θ

Rearranging the equation, we have:

cos²θ = 1 - 3 = -2

Since cos θ cannot be negative, this means that there is no real value for cos θ. Hence, we cannot determine the exact value of r.

b. To find the cosecant (csc) θ, we can use the reciprocal relationship between sin and csc:

csc θ = 1 / sin θ

Substituting the value of sin θ from equation (3), we have:

csc θ = 1 / (-√3) = -1/√3 = -√3/3

c. Since we were unable to determine the exact value of cosine (cos) θ, we cannot find its exact value.

d. To find the cotangent (cot) θ, we can use the reciprocal relationship between tan and cot:

cot θ = 1 / tan θ

Substituting the given value of tan θ, we have:

cot θ = 1 / (-√3/8) = -8/√3 = -(8√3)/3

Therefore, the exact values are:
a. r = N/A
b. csc θ = -√3/3
c. cos θ = N/A
d. cot θ = -(8√3)/3