If the torque required to loosen a nut that holds a wheel on a car has a magnitude of 55 n·m, what force must be exerted at the end of a 0.47 m lug wrench to loosen the nut when the angle is 48°?

Either 175 N or 157 N depending upon how the value of 48° was measured from.

You didn't mention if the angle of 48° is from the lug wrench itself, or if it's from the normal to the lug wrench. So I'll solve for both cases and you'll need to select the desired answer.

Since we need a torque of 55 N·m to loosen the nut and our lug wrench is 0.47 m long, that means that we need 55 N·m / 0.47 m = 117 N of usefully applied force in order to loosen the nut. This figure will be used for both possible angles.

Ideally, the force will have a 0° degree difference from the normal and 100% of the force will be usefully applied. Any value greater than 0° will have the exerted force reduced by the cosine of the angle from the normal. Hence the term "cosine loss".

If the angle of 48° is from the normal to the lug wrench, the usefully applied power will be:

U = F*cos (48)

where

U = Useful force

F = Force applied

So solving for F and calculating gives:

U = F*cos (48)

U/cos (48) = F

117 N/0.669130606 = F

174.8537563 N = F

So 175 Newtons of force is required in this situation.

If the 48° is from the lug wrench itself, that means that the force is 90° - 48° = 42° from the normal. So doing the calculation again (this time from where we started plugging in values) we get

U/cos (42) = F

117/0.743144825 = F

157.4390294 = F

Or 157 Newtons is required for this case.