Indeed, they were. Agincourt actually comes quite late in the timeline of English archery in warfare, and arguably isn't the most effective use of archers, though certainly more well known and cited thanks in no small part to Shakespeare.

The legacy of English archery is cemented in the Assize of Arms 1252, which is the law that mandated the ownership of bows and arrows by the population and the practice of archery on Sundays after church. This would provide England with a large pool of archers that could be recruited for campaigns against the Scots and the French.

A brief timeline of notable battles in which English archers played a substantial role:

• Falkirk, 1298: The English archers were extremely effective against the unsupported, immobile Scottish schiltrons, leading to the defeat of William Wallace.
• Halidon Hill, 1333: English archers take position on a hill and shoot down into the Scottish pikemen downhill. The Scottish infantry, forced to attack due to the superiority of English archers, are destroyed as they attempt to charge uphill.
• Crecy, 1346: Perhaps the most "textbook" example of English archery in the Hundred Years' War. The English archers outmatched the Genoese crossbowmen, who were hampered by damp strings and did not have their pavise shields. The French were under constant barrage from the English archers, leading to a comprehensive defeat that would be unmatched until Agincourt.
• Neville's Cross, 1346: Perhaps more a testament to the ready availability of trained archers, this battle against the Scots in northern England relied on the recruitment of archers from Lancashire and other northern counties. The heavily armoured Scottish infantry are more resistant to the archers, but King David II is hit in the face. The English archers flank the Scottish reinforcements and were effective against these lighter troops.
• Poitiers, 1356: Like Crecy, the English archers were deployed in "textbook" fashion and harassed the French troops until they ran out of arrows. Notably, the battle featured mounted archers - English archers who rode behind the enemy flank before dismounting and fighting.
• Homildon Hill, 1402: English archers dislodge an army of Scots who have the advantage of high ground.

Agincourt, in 1415, is often regarded as the high water mark of English archery, though perhaps the least typical deployment of archers. Normally English archers fought from prepared positions and baited the enemy into attacking. At Agincourt, the English were forced to open the battle by moving the archers forward into range. The desperation of the battle also forced the archers to fight in the melee, which was enough to turn the battle, though a rather exceptional case.

Subsequent battles would see the decline of English archery, partly due to the diminishing numbers of trained archers after decades of conflict. Patay (1429) was the antithesis of Agincourt - the archers were caught out of position and were ridden down. Formigny (1450) saw another defeat, with the English archers goaded into attack by French guns and then dispersed by French cavalry. Archers continued to be used by both sides in the Wars of the Roses, and the final recorded deployment English archers was during the English Civil War.

Sources

• Loades, Mike. The Longbow
• Featherstone, Donald. The Bowmen of England