To find the values of sin ∅, tan ∅, and to determine the angle ∅ in both degrees and radians, we can use the given value of cos ∅.
Given: cos ∅ = -√3/2 and π < ∅ < 3π/2
We know that cos ∅ = adjacent/hypotenuse, and since cos ∅ = -√3/2, we can assign values to the adjacent and hypotenuse sides of a right triangle. Let's assume the adjacent side is -√3 and the hypotenuse is 2.
Using the Pythagorean theorem, we can find the value of the opposite side of the triangle:
opposite^2 = hypotenuse^2 - adjacent^2
opposite^2 = 2^2 - (-√3)^2
opposite^2 = 4 - 3
opposite^2 = 1
opposite = 1
Now we have the values for the three sides of the right triangle: opposite = 1, adjacent = -√3, and hypotenuse = 2.
Using these values, we can determine the values of sin ∅ and tan ∅:
sin ∅ = opposite/hypotenuse = 1/2 = 0.5
tan ∅ = opposite/adjacent = 1/(-√3) = -1/√3
Now, let's find the value of ∅ in both degrees and radians:
We know that π < ∅ < 3π/2, which means ∅ lies in the second quadrant. In the second quadrant, sin ∅ is positive, and cos ∅ is negative.
Since cos ∅ = -√3/2, we can use the cosine inverse (arccos) function to find the angle ∅:
∅ = arccos(-√3/2)
∅ ≈ 150.52 degrees
To convert ∅ to radians, we can use the conversion factor π/180:
∅ in radians = ∅ in degrees * (π/180)
∅ in radians ≈ 150.52 * (π/180)
∅ in radians ≈ (5π/6) radians
So, the values are:
sin ∅ ≈ 0.5
tan ∅ ≈ -1/√3
∅ ≈ 150.52 degrees
∅ ≈ (5π/6) radians