Problem 8. 5 pts. Find the exact values of the remaining 5 circular functions for angle θ,

given that sinθ = 1/9 and θ is an angle in the 2
nd quadrant.

sinØ = y/r = 1/9 ---> y = 1, r = 9

Make a sketch of the triangle in quad II
so x^2 + y^2 = 9^2
x^2 + 1 = 81
x^2 = 80
x = ±√80 = ±4√5
but we are in II, so x = -4√5

sinØ = 1/9 ---> the given
cosØ = -4√5/9
tanØ = 1/(-4√5) or - √5/20

I am sure you can take over and finish

8, find the exact values of the five remaining trigonometric functions for the acute angle 𝜽

To find the exact values of the remaining five circular functions (cosine, tangent, cotangent, secant, and cosecant) for angle θ, we will use the given information that sinθ = 1/9 and the fact that θ is an angle in the 2nd quadrant.

First, let's find the value of cosine (cosθ):
Since θ is in the 2nd quadrant, cosine is negative in this quadrant. We can use the Pythagorean identity to find the value of cosine:
cos²θ + sin²θ = 1
cos²θ + (1/9)² = 1
cos²θ + 1/81 = 1
cos²θ = 1 - 1/81
cos²θ = 80/81
Taking the square root of both sides, we get:
cosθ = ±√(80/81)
Since θ is in the 2nd quadrant, cosine is negative, so we take the negative square root:
cosθ = -√(80/81)
Simplifying the square root, we get:
cosθ = -√(16·5)/(9·9)
cosθ = - (4√5)/(9)

Next, let's find the value of tangent (tanθ):
The tangent is defined as the ratio of sine to cosine:
tanθ = sinθ / cosθ
Substituting the known values, we get:
tanθ = (1/9) / (- (4√5)/(9))
To divide fractions, we invert the divisor and multiply:
tanθ = (1/9) * (-9/(4√5))
Simplifying, we get:
tanθ = -1/(4√5)

Now, let's find the value of cotangent (cotθ):
cotθ is the reciprocal of tangent, so we can find it by taking the reciprocal of the tangent value we just found:
cotθ = 1 / tanθ
cotθ = 1 / (-1/(4√5))
To divide fractions, we invert the divisor and multiply:
cotθ = 1 * (-4√5/1)
Simplifying, we get:
cotθ = -4√5

Next, let's find the value of secant (secθ):
secθ is the reciprocal of cosine, so we can find it by taking the reciprocal of the cosine value we found earlier:
secθ = 1 / cosθ
secθ = 1 / (-(4√5)/(9))
To divide fractions, we invert the divisor and multiply:
secθ = 1 * (9/(-(4√5))
Simplifying, we get:
secθ = -9/(4√5)

Finally, let's find the value of cosecant (cscθ):
cosecθ is the reciprocal of sine, so we can find it by taking the reciprocal of the sine value given:
cosecθ = 1 / sinθ
cosecθ = 1 / (1/9)
To divide fractions, we invert the divisor and multiply:
cosecθ = 1 * (9/1)
Simplifying, we get:
cosecθ = 9

Therefore, the exact values of the remaining five circular functions for angle θ, given that sinθ = 1/9 and θ is an angle in the 2nd quadrant, are:
cosθ = -(4√5)/(9)
tanθ = -1/(4√5)
cotθ = -4√5
secθ = -9/(4√5)
cosecθ = 9.