The Law of Cosines is used for this purpose.
c² = a² + b² - 2ab·cos(C)
c² = 5² + 4² - 2·5·4·cos(60°)
c² = 25 + 16 - 20 = 21
c = √21
The length of side c is √21 ≈ 4.583.
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If you really intend the side lengths to be 55 and 44, your result is c≈50.408, the above value times 11.