To find the value of the cotangent of the acute angle θ given that sin(θ) = 8/9, we can follow these steps:
1. **Understand the definition of cotangent**: The cotangent of an angle θ is the reciprocal of the tangent. Therefore, we can express cot(θ) as:
cot(θ) = cos(θ) / sin(θ)
2. **Use the Pythagorean identity**: We know from the Pythagorean identity that:
sin²(θ) + cos²(θ) = 1
Given that sin(θ) = 8/9, we can substitute:
(8/9)² + cos²(θ) = 1
Calculating the left side:
64/81 + cos²(θ) = 1
To find cos²(θ), we subtract 64/81 from 1:
cos²(θ) = 1 – 64/81 = 81/81 – 64/81 = 17/81
3. **Find cos(θ)**: Since θ is an acute angle, cos(θ) will be positive:
cos(θ) = √(17/81) = √17 / 9
4. **Plug values into cot(θ)**: Now that we have sin(θ) and cos(θ), we can find cot(θ):
cot(θ) = cos(θ) / sin(θ) = (√17 / 9) / (8 / 9)
This simplifies to:
cot(θ) = √17 / 8
Thus, the value of cotangent for the acute angle θ is:
cot(θ) = √17 / 8
In conclusion, when you know the sine of an acute angle in a right triangle, it’s straightforward to calculate its cotangent through these relationships!