Trigonometry/Some preliminary results

We prove some results that are needed in the application of calculus to trigonometry.

Reference Image for Proof

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

.