Trigonometry/Some preliminary results

We prove some results that are needed in the application of calculus to trigonometry. thumb|Reference Image for Proof|433x433px Theorem: If ; is a positive angle, less than a right angle (expressed in radians), then . Proof: Consider a circle, centre , radius , and choose two points and on the circumference such that is less than a right angle. Draw a tangent to the circle at , and let produced intersect it at . Clearly i.e. and the result follows. Corollary: If is a negative angle, more than minus a right angle (expressed in radians), then . [This follows from and .] Corollary: If is a non-zero angle, less than a right angle but more than minus a right angle (expressed in radians), then . Theorem: As and . Proof: Dividing the result of the previous theorem by and taking reciprocals, . But tends to as tends to , so the first part follows. Dividing the result of the previous theorem by and taking reciprocals, . Again, tends to as tends to , so the second part follows. Theorem: If is as before, then . Proof: . Theorem: If is as before, then . Proof: . . . Theorem: and are continuous functions. Proof: For any , , since cannot exceed and cannot exceed . Thus, as , proving continuity. The proof for cos(θ) is similar, or it follows from .