In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. If we have an acute angle θ such that sin(θ) = 6/7, we can use this relationship to find the value of cotangent (cot) for the angle.
The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side, or cot(θ) = cos(θ) / sin(θ). To find cot(θ), we need to first determine the values of sin(θ) and cos(θ).
Since we already know sin(θ) = 6/7, we can find cos(θ) using the Pythagorean identity:
sin²(θ) + cos²(θ) = 1
Substituting in our known value for sin(θ):
(6/7)² + cos²(θ) = 1
This simplifies to:
36/49 + cos²(θ) = 1
Next, we rearrange it to solve for cos²(θ):
cos²(θ) = 1 – 36/49
cos²(θ) = 49/49 – 36/49 = 13/49
Taking the square root gives us:
cos(θ) = √(13/49) = √13 / 7
Now that we have both sine and cosine values, we can calculate the cotangent:
cot(θ) = cos(θ) / sin(θ)
Substituting in the values we found:
cot(θ) = (√13 / 7) / (6/7)
This dramatically simplifies down to:
cot(θ) = √13 / 6
Thus, the value of cotangent for the acute angle in the right triangle is:
cot(θ) = √13 / 6