If \cot \left(\sin^{-1} \frac{1}{5}\right) = m, what is the value of m^2?

cot(arcsin(1/5))

draw a triangle. If sin=1/5, cot=√24
m^2=24

To find the value of m^2, let's first simplify the given expression step by step.

The expression \cot \left(\sin^{-1} \frac{1}{5}\right) represents the cotangent of the angle whose sine is 1/5.

Step 1: Find the angle whose sine is 1/5
The sine function takes an angle as input and gives the ratio of the opposite side to the hypotenuse in a right triangle with that angle. Since we know that the sine of an angle is 1/5, we can calculate the opposite side length as 1 and the hypotenuse length as 5. This is a Pythagorean triple (3-4-5).

So, the angle whose sine is 1/5 is the angle whose opposite side is 1 and hypotenuse is 5.

Step 2: Find the cotangent of the angle
The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle. In our case, we need to find the cotangent of the angle whose opposite side is 1 and adjacent side is 3 (since the hypotenuse is 5).

Using the definition of cotangent, we have:
\cot \left(\sin^{-1} \frac{1}{5}\right) = \frac{\text{adjacent side}}{\text{opposite side}} = \frac{3}{1} = 3.

So, we have found that m = 3.

Step 3: Calculate the value of m^2
To find m^2, we simply square the value of m:
m^2 = (3)^2 = 9.

Therefore, the value of m^2 is 9.