Answer:
(6√13)/13
Explanation:
You want the exact value of cot(arcsin(√13/7)).
Trig ratios
The relevant trig ratios are ...
Sin = Opposite/Hypotenuse
Tan = Opposite/Adjacent
Cot = 1/Tan = Adjacent/Opposite
Pythagorean theorem
The Pythagorean theorem can be used to find the side adjacent to the angle whose sine is √13/7. Using the sine ratio, we can take the opposite side to be √13, and the hypotenuse to be 7. Then the adjacent side is ...
adjacent² +opposite² = hypotenuse²
adjacent² +(√13)² = 7²
adjacent² = 49 -13 = 36
adjacent = √36 = 6
Cotangent
Then the cotangent of the angle is ...
cot(arcsin(√13/7)) = adjacent/opposite = 6/√13
cot = (6√13)/13