In this thesis we construct rogue waves occurring in the sine-Gordon equation. An algebraic method is used to find explicit solutions to a Lax pair of equations. The Lax pair being studied is compatible with solutions to the sine-Gordon equation. Rotational and librational traveling wave solutions to the sine-Gordon equation are considered in the Lax pair. The Darboux transformation is applied with the Lax pair solutions computed at the rotational and librational waves to generate algebraic solitons and rogue waves, respectively. The rogue waves occur on the end points of the Floquet-Lax spectrum bands and can achieve a magnification factor of at most 3. / Thesis / Master of Science (MSc)