It says in Vayikra (32:42) that "in the seventh month on the first day you shall have a rest day...and you shall make a remembrance with the shofar." Traditionally, this day of Rosh Hashana has become a day of reflection, a day of repentance, the first of the ten days on which our fate is determined and sealed by Hashem. The shofar has therefore become a symbol of repentance, stemming from the fact that it is first used in Elul, the introductory month of Teshuvah, as a call to the Jewish people to do Teshuvah. So ultimately, the day and its symbols are viewed on a somber note. One wonders then, why should they be linked to יום תרועה, literally a day of joyful noise?
Of course, no one doubts the great importance of the shofar, for as it says in the Rambam's יד החזקה (הל' שופר א:ב), "In the Beit Hamikdash on Rosh Hashana they blew one shofar and two silver חצוצרות, silver trumpets, one on each side of the shofar. The Shofar was long and the חצוצרות were short," as not to take away from the Kedusha of the day as represented by the Shofar.
But still the question of the joy remains. The history of the Shofar can be traced back to the Akeida, as it says (בראשית כב:יג) "And Avraham raised his eyes and behold! A ram." That ram is represented by a shofar, a ram's horn, at a time of great joy for Avraham as he realized that he would not need to sacrifice his son Yitzchak.
The Shofar can also be traced to the construction to the time of Matan Torah when, according to Midrash, the Shofar was sounded. This also was a time of great joy for Bnai Yisrael. So now the Shofar has been linked to joyous occasions. But why is it used at such a somber time and why is that day connected with this joy?
The Sforno says that the answer can be found in Tehillim (18). It's written there that "we should sing aloud to Hashem, our Strength, make a joyful noise." Later on in that פרק it says, "Blow the Shofar at the new moon... of our festival for it is an ordinance from the time of Yaakov." Many may recognize the above phrase of תקעו בחודש שופר which is the introduction to the Amidah of Maariv on Rosh Hashana. The reason that these Pesukim from Tehillim are connected is that at this time of the New Year we rejoice because Hashem is treating and judging us favorably, so it is only appropriate that we should want to offer up a joyful noise, for it says in Yishayahu (33:22), כי ה` שופטנו...הוא יושענו , "For Hashem is our judge... he will save us."