Explanation:
1) We can start by finding the adjacent and opposite sides of the right triangle using the Pythagorean theorem. Let x be the length of the adjacent side, then:
tan θ = (3-√3)/9
tan θ = opposite/adjacent
(3-√3)/9 = opposite/x
opposite = (3-√3)x/9
Using Pythagorean theorem:
(adjacent)^2 + (opposite)^2 = (hypotenuse)^2
x^2 + [(3-√3)x/9]^2 = (hypotenuse)^2
x^2 + (9-6√3+3)x^2/81 = (hypotenuse)^2
(82-54√3)x^2/81 = (hypotenuse)^2
We can simplify this expression to find the hypotenuse:
(hypotenuse)^2 = [(82-54√3)/81]x^2
hypotenuse = sqrt[(82-54√3)/81]x
Now we can find the other trig ratios:
cos θ = adjacent/hypotenuse = x/sqrt[(82-54√3)/81]x
cos θ = 1/sqrt[(82-54√3)/81]
cos θ = sqrt[(82+54√3)/81] / [(82-54√3)/81]
cos θ = sqrt(82+54√3) / (82-54√3)
sin θ = opposite/hypotenuse = (3-√3)x/9 / sqrt[(82-54√3)/81]x
sin θ = (3-√3) / sqrt(82-54√3)
2) To find the other trig ratios, we need to first find the adjacent and hypotenuse sides of the right triangle. Let x be the length of the adjacent side, then:
sin θ = opposite/hypotenuse = 8/10
opposite = (8/10)hypotenuse = 0.8hypotenuse
Using Pythagorean theorem:
(adjacent)^2 + (opposite)^2 = (hypotenuse)^2
x^2 + (0.8hypotenuse)^2 = hypotenuse^2
x^2 + 0.64h^2 = h^2
x^2 = 0.36h^2
x = 0.6h
Now we can find the other trig ratios:
cos θ = adjacent/hypotenuse = 0.6h/h = 0.6
tan θ = opposite/adjacent = (8/10)/0.6 = 4/3
cot θ = 1/tan θ = 3/4