Review of the history of trigonometry content and pedagogy indicates the necessity and importance of trigonometry in the school curriculum (e.g., van Brummelen, 2009; van Sickel, 2011). For example, understanding trigonometric functions is a requirement for understanding some other areas of science, such as Newtonian physics, architecture, surveying, and several branches of engineering. Research indicates that teachers have a narrow and inadequate understanding of trigonometry, researchers (e.g., Hertel, 2013; van Sickel, 2011; Weber, 2005) posited that learning trigonometry in a way that fosters and advances quantitative reasoning can help alleviate difficulties faced in the teaching of trigonometry. Examining participants' reasoning is one way of determining if an instruction sequence promotes the desired understanding (Hiebert, 2003; Tall, 1996). Missing from the literature is investigative work that uncovers what kind of reasoning teachers use when engaged in a trigonometry instruction sequence that promotes quantitative reasoning. In this dissertation, I will examine pre-service secondary mathematics teachers' reasoning about trigonometric functions when an instructional sequence (Hertel & Cullen, 2011) of trigonometric activities was used. Ultimately, this research is intended to shine more light on how a particular approach (line-segment) can influence prospective secondary mathematics teachers' knowledge of trigonometric functions and will seek to benefit the pre-service teachers by developing a strong comprehension of trigonometric functions. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]