The 75 mm gun on the Sherman was already reasonably effective against the Panzer IV and the various vehicles that were based off of it, being able to kill them at normal combat ranges. A post-war study revealed that the average distance that a US tank killed a German tank was 893 yards, while the average distance that a German tank killed a US tank was 943 yards. Another study also proved that the Sherman was markedly superior to the Panther in most tactical situations (the engagements studied); an average numerical advantage of 1.2:1, 1.1 times as effective when on the defensive, 8.4 times as effective when on the offensive, and 3.6 times as effective overall. When it encountered short-barreled Panzer IVs and long-barreled Panzer IIIs in North Africa, the 75 mm gun of the Sherman proved superior. It was unable, however, to penetrate the frontal armor of the few Tiger Is it encountered, but it could knock them out from the side or rear. The US Army, besides the campaigns in North Africa and Italy, encountered relatively few Tiger Is, their main opponents being, in order of commonality, the Panzer IV, Panther, and King Tiger (the most common English translations of königstiger, "king tiger" and "royal tiger" are technically incorrect; könig does by itself mean "king", but the word königstiger, in German, is used to refer to the Bengal tiger) I usually try to avoid pedantic gun-penetration-vs-armor-thickness-at-this-time-of-day-or-humidity kinds of things, but here I'll lay out some charts just for reference;

Gun	Shot	Angle	457 m	914 m	1,371 m
75 mm M2 L/31 (US)	APCBC	30 deg	60 mm	53 mm	46 mm
75 mm M3 L/40 (US)	APCBC	30 deg	66 mm	60 mm	55 mm
76 mm M1 L/52 (US)	APCBC	30 deg	93 mm	88 mm	82 mm
3-inch M7 L/50 (US)	APCBC	30 deg	93 mm	88 mm	82 mm
7,5 cm KwK/StuK 40 L/43 (German)	APCBC	30 deg	91 mm	81 mm	72 mm
7,5 cm KwK/StuK 40 L/48 (German)	APCBC	30 deg	96 mm	85 mm	74 mm
7,5 cm Kwk/StuK 42 L/70 (German)	APCBC	30 deg	124 mm	111 mm	99 mm
8,8 cm Kwk 36 L/56 (German)	APCBC	30 deg	110 mm	100 mm	91 mm
8,8 cm Kwk 43 L/71 (German)	APCBC	30 deg	185 mm	165 mm	148 mm

75 mm M3 APCBC M61 vs Panzer IV, Panther, Tiger, and King Tiger:

Tank	Max assured pen. distance
Panzer IV Ausf J hull front	~1,200 m
Pz IV Ausf J hull side	~2,000+ m
Pz IV Ausf J hull rear	~2,000+ m
Pz IV Ausf J turret front	~2,000+ m
Pz IV Ausf J turret side	~2,000+ m
Pz IV Ausf J turret rear	~2,000+ m
Panzer V Ausf G hull front	Unable to pen.
Pz V Ausf G hull side	~1,900 m
Pz V Ausf G hull rear	~1,900 m
Pz V Ausf G turret front	Unable to pen.
Pz V Ausf G turret side	~2,000+ m
Pz V Ausf G turret rear	~2,000+ m
Panzer VI Ausf E (Tiger) hull front	Unable to pen.
PzVI Ausf E hull side	~650 m
Pz VI Ausf E hull rear	~650 m
Pz VI Ausf E turret front	Unable to pen.
Pz VI Ausf E turret side	~650 m
Pz VI Ausf E turret rear	~650 m
Panzer VI Ausf B (King Tiger) hull front	Unable to pen.
Pz VI Ausf B hull side	~100 m
Pz VI Ausf B hull rear	~100 m
Pz VI Ausf B turret front	Unable to pen.
Pz VI Ausf B turret side	~200 m
Pz VI Ausf B turret rear	~200 m

76 mm M1 and 3-inch M7 APCBC M62 vs Panzer IV, Panther, Tiger, and King Tiger:

Tank	Max assured pen. distance
Panzer IV Ausf J hull front	~2000+ m
Pz IV Ausf J hull side	~2000+ m
Pz IV Ausf J hull rear	~2000+ m
Pz IV Ausf J turret front	~2000+ m
Pz IV Ausf J turret side	~2000+ m
Pz IV Ausf J turret rear	~2000+ m
Panzer V Ausf G hull front	Unable to pen.
Pz V Ausf G hull side	~2,000+ m
Pz V Ausf G hull rear	~2,000+ m
Pz V Ausf G turret front	~1,400 m (based on armor thicknesses and pen. values only)
Pz V Ausf G turret side	~2,000+ m
Pz V Ausf G turret rear	~2,000+ m
Panzer VI Ausf E (Tiger) hull front	~1,100 m
Pz VI Ausf E hull side	~2,000+ m
Pz VI Ausf E hull rear	~2,000+ m
Pz VI Ausf E turret front	~1,400 m
Pz VI Ausf E turret side	~2,000+ m
Pz VI Ausf E turret rear	~2,000+ m
Panzer VI Ausf B (King Tiger) hull front	Unable to pen.
Pz VI Ausf B hull side	~1,600 m
Pz VI Ausf B hull rear	~1,600 m
Pz VI Ausf B turret front	Unable to pen.
Pz VI Ausf B turret side	~1,800 m
Pz VI Ausf B turret rear	~1,800 m

There is a phenomenon which takes into account the properties of rounds and contact with armor, known as "shatter gap". If a round is too "soft" based upon its Rockwell measure, (most US and Soviet rounds fell into this category, but most British and German rounds didn't) its velocity is increased, and armor thickness is held constant, there occurs a point where the projectile nose begins absorbing too much of the impact energy and fails, essentially "exploding" on contact. For APCBC shot, this point occurs when the ratio of projectile penetration/armor resistance (true or effective thickness) is between 1.05 and 1.25, and projectiles are traveling faster than 2,000 fps; for AP shot, the upper and lower limits are a bit higher. Below the range, probability curves apply; above it, all hits will penetrate. This means that projectiles can refuse to penetrate at shorter ranges, but successfully penetrate at longer ranges.

Remember that you don't need to penetrate a tank's armor in order to kill it. The Panther and King Tiger were overly complicated and suffered from serious engine and transmission issues, and the Panzer IV was originally an 18 ton tank that by its terminus was up to 25 tons; a broken down tank is just as dead as a blasted one. US tank crews would often fire at the track, breaking it, and then hit the enemy tank with a white phosphorus or smoke shell; the smoke would get sucked inside the tank, irritating or injuring the crew and forcing them to abandon the vehicle. The 75 and 76 mm (as well as the 3-inch, which had identical performance to the 76 mm gun) guns did not generally have any trouble with the armor of the Panzer IV, which never exceeded 80 mm of flat plate on any tank variant; the 76 mm M1 was the rough equal of the 7,5 cm KwK 40 L/43 and L/48. The thick and steeply-sloped glacis plate (80 mm at 55 degrees, giving an effective head-on thickness of 145 mm!) of the Panther was all but immune to the 75 mm gun of the Sherman; the Tiger I was also immune, and the King Tiger doesn't even need to be mentioned! Jentz and Doyle cannot find one combat instance in which the glacis plate of the King Tiger was penetrated. The 76 mm gun actually had a very slim chance of penetrating the glacis of the Panther, but only at point-blank range, at a very favorable angle, and firing HVAP. The Tiger I was a different story; the Tigerfibel (Tiger crew's manual) frowned against engaging enemy tanks within 500 m, and the 76 mm gun could penetrate the Tiger's front plate out past 1,000 m. In this respect there was no change, as the 75 mm and 76 mm guns generally couldn't, in "combat" situations, handle the Panther or King Tiger (remember, Tiger Is were very uncommon) from the front at any angle, The first 76 mm-gunned Shermans that arrived in the UK in early 1944 for tests were held back as they were "unneeded", under the assumption that the 75 mm gun could handle anything that it was most likely to see, and that the Panther was a heavy tank that would be seen only in small numbers like the Tiger. When, in Normandy, the 75 mm gun proved to be ineffective (shortly afterwards, the 3-inch gun did as well) against large numbers of Panthers, the 76 mm tanks began to be distributed widely.

Even the 90 mm gun of the M26 Pershing had problems with the Panther at certain angles, and a special shell was developed to handle it. Both the Panther and Tiger could be easily killed from the side or rear at normal combat ranges, as they had thin (40-60 mm) side armor and poor methods of stowing ammunition (much like that of the early Shermans) To provide a counter to the perceived dominance of German heavy tanks, a study performed by Wa Pruef 1 (essentially one of the many German equivalents to Aberdeen Proving Ground) in October 1944 found that when a Sherman (56-degree glacis) was angled at 30 degrees, the glacis armor was impenetrable to the 7,5 cm KwK 40 from any range, while the 75 mm M3 could penetrate the Panzer IV from 100 m in the same situation. A favorite tactic of Sherman crews was to bounce a shell off of the lower part of the rounded mantlet (the mantlet of the Panther was immune to penetration by the 75 mm gun, and the 76 mm gun could only penetrate without the shell ricocheting off the rounded surface at point-blank range, less than 200 m) of the Panther, penetrating the thin roof armor and potentially detonating the rounds stowed behind the driver and assistant driver, or aiming for the machine gun ball mount. In Italy, tank destroyer crews did something similar ricocheting shells off the ground and into the bellies of Elefant tank destroyers. The Panther Ausf. G incorporated a mantlet "chin" to stop this tactic from being effective, and eliminated the driver's vision flap. The quality of armor in these late German tanks due to poor workmanship and a lack of critical materials was often not very good; repeated hits in the same place or with heavy explosive shells would cause spalling, the bursting of welds, or the armor cracking and caving in.

Sources:

Gun pen. tables

Gun pen. calculator

Tigerfibel

WWII Ballistics: Armor and Gunnery, by L. Bird and R. Livingston

Data on World War II Tank Engagements: Involving the US Third and Fourth Armored Divisions, by D.C. Hardison

Survey of Allied Tank Casualties in World War II, by A.D. Coox and L.V. Naisawald

Panzerkampfwagen IV Ausf. G, H and J 1942-45, by T. Jentz and H. Doyle

Armored Thunderbolt: The US Army Sherman in World War II, by S.J. Zaloga

Panther vs Sherman: Battle of the Bulge 1944, by S.J. Zaloga

Panther: Germany's Quest for Combat Dominance, by M. Green

M4 (76 mm) Sherman Tank 1943-65, by S.J. Zaloga

M26/M46 Pershing Tank 1943-53, by S.J. Zaloga