How do you explain the difference between using the trigonometric ratios to solve for a missing angle in a right triangle versus using the reciprocal ratios?

In the trigonometric identities, secant is the inverse of cosine, cosecant is the inverse of sine function and lastly, cotangent is the inverse function of tangent. in this case, while the ratio of sine is opposite over hypotenuse, cosecant has the ratio that is reciprocal to that of sine function, that is opposite over opposite. This applies to all other functions. The difference would be the orientation and location of the angle of reference making use of the SOH CAH TOA

you want to first identify your angle of reference

then whatever is across that angle would be the opposite side

if you were given an angle, the opposite side, you can solve for the hypotenuse

using Sin (angle) = opposite side/h

then solve for h using algebraic manipulation the adjacent side is the side next to the angle of reference. it is usually perpendicular to the opposite side

if you were given the length of the adjacent side and opposite side you can solve for the angle

you can use TOA

tangent (angle) = opposite/adjacent

then obtain the arctan of the resulting ratio