To find the value of cot(θ) when sin(θ) = 8/9, we can use the relationship between the trigonometric functions. We know that cot(θ) is the reciprocal of tan(θ), and tan(θ) can be determined using the sine and cosine functions.
Given that sin(θ) = 8/9, we can find cos(θ) using the Pythagorean identity:
sin²(θ) + cos²(θ) = 1.
Substituting sin(θ):
(8/9)² + cos²(θ) = 1
64/81 + cos²(θ) = 1
To solve for cos²(θ), we subtract 64/81 from 1:
cos²(θ) = 1 – 64/81 = 17/81.
Next, we take the square root to find cos(θ):
cos(θ) = √(17/81) = √17/9.
Now we can find tan(θ):
tan(θ) = sin(θ)/cos(θ) = (8/9) / (√17/9) = 8/√17.
Finally, we get cot(θ), which is the reciprocal of tan(θ):
cot(θ) = 1/tan(θ) = √17/8.
Thus, the value of cot(θ) is:
√17/8.