For a more thorough discussion of these functions along the lines given here, see my pre calculus book published by Worldwide Center of Math. For a non geometric treatment, see my book Single variable advanced calculus or for a different way, Pure Mathematics by Hardy [19]. I much prefer methods which do not depend on plane geometry because with this approach, many of the most difficult and unpleasant considerations become obvious and then one can use the machinery of calculus to discuss geometric significance instead of relying so much on axioms from geometry which may or may not be well remembered. However, I am giving the traditional development of this subject here.
An angle consists of two lines emanating from a point as described in the following picture. How can angles be measured? This will be done by considering arcs on a circle. To see how this will be done, let θ denote an angle and place the vertex of this angle at the center of the circle. Next, extend its two sides till they intersect the circle. Note the angle could be opening in any of infinitely many different directions. Thus this procedure could yield any of infinitely many different circular arcs. Each of these arcs is said to subtend the angle.
Take an angle and place its vertex (the point) at the center of a circle of radius r. Then, extending the sides of the angle if necessary till they intersect the circle, this determines an arc on the circle which subtends the angle. If r were changed to R, this really amounts to a change of units of length. Think, for example, of keeping the numbers the same but changing centimeters to meters in order to produce an enlarged version of the same picture. Thus the picture looks exactly the same, only larger. It is reasonable to suppose, based on this reasoning that the way to measure the angle is to take the length of the arc subtended in whatever units being used and divide this length by the radius measured in the same units, thus obtaining a number which is independent of the units of length used, just as the angle itself is independent of units of length. After all, it is the same angle regardless of how far its sides are extended. This is how to define the radian measure of an angle and the definition is welldefined. Thus, in particular, the ratio between the circumference (length) of a circle and its radius is a constant which is independent of the radius of the circle^{1} . Since the time of Euler in the 1700’s, this constant has been denoted by 2π. In summary, if θ is the radian measure of an angle, the length of the arc subtended by the angle on a circle of radius r is rθ.
So how do we obtain the length of the subtended arc? For now, imagine taking a string, placing one end of it on one end of the circular arc and then wrapping the string till you reach the other end of the arc. Stretching this string out and measuring it would then give you the length of the arc. Later a more precise way of finding lengths of curves will be given.
Definition 2.3.1 Let A be an angle. Draw a circle centered at A which intersects both sides of the angle. The radian measure of the angle is the length of this arc divided by the radius of the circle.
Thus the radian measure of A is l∕r in the above. (Note that the radian measure of an angle does not depend on units of length. )There is also the wrong way of measuring angles. In this way, one degree consists of an angle which subtends an arc which goes 1∕360 of the way around the circle. The measure of the angle consists of the number of degrees which correspond to the given angle.
We avoid the wrong way of measuring angles in calculus. This is because all the theorems about the circular functions having to do with calculus topics involve the angle being given in radians.
In any triangle, the sum of the radian measures of the angles equals π. Lets review why this is so.
Consider the following picture.
The line at the top is chosen to be parallel to the line on the base. Then from axioms of geometry about alternate interior angles, the diagram is correctly labeled. Now if you consider the angle formed by a point on a straight line, then it is obvious that the circle centered at this point has exactly half of it subtended by the line. Thus the radian measure of the angle is π. If we identify the radian measure of each of these angles with the label used for the angle, it follows that the sum of the measures of the angles of the triangle, α + β + γ equals π.
The following proof of the Pythagorean theorem is due to Euclid a few hundred years B.C. A right triangle is one in which one of the angles has radian measure π∕2. It is called a right triangle. Thus if this angle is placed with its vertex at (0,0) its sides subtend an arc of length π∕2 on the unit circle. The hypotenuse is by definition the side of the right triangle which is opposite the right angle. From the above observation, both of the other angles in a right triangle have radian measure less than π∕2
Theorem 2.3.2 (Pythagoras) In a right triangle the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
Proof: Consider the following picture in which the large triangle is a right triangle and D is the point where the line through C perpendicular to the line from A to B intersects the line from A to B. Then c is defined to be the length of the line from A to B, a is the length of the line from B to C, and b is the length of the line from A to C. Denote by DB the length of the line from D to B.
Then δ + γ = π∕2 and β + γ = π∕2. Therefore, δ = β. Also from this same theorem, α + δ = π∕2 and so α = γ. Therefore, the three triangles shown in the picture are all similar because they have the same angles at vertices. From the similar triangle axiom in geometry, the corresponding parts are proportional. Then

Therefore, cDB = a^{2} and

so
c^{2}  = cDB + b^{2} 
= a^{2} + b^{2}. 
This proves the Pythagorean theorem. ^{2} ■
This theorem implies there should exist some such number which deserves to be called
Definition 2.3.3 Let t ∈ ℝ. Then p
Points in the plane may be identified by giving a pair of numbers. Suppose there are two points in the plane and it is desired to find the distance between them. There are actually many ways used to measure this distance but the best way, and the only way used in this book is determined by the Pythagorean theorem. Consider the following picture.
In this picture, the distance between the points denoted by

or even

and it would make no difference in the resulting number. The distance between the two points is written as
Thus, given an x and y axis at right angles to each other in the usual way, a relation which describes a point on the circle of radius 1 which has center at
Given a real number t ∈ ℝ, I will describe a point p
Definition 2.3.4 Let t ∈ ℝ. If t is positive, take a string of length t, place one end at the point
Definition 2.3.5 Let t ∈ ℝ. Then p
We say that
The other functions from trigonometry are defined in the usual way

Since
 (2.2) 
This is the most fundamental identity in trigonometry. Also, directly from the definition it follows that
 (2.3) 
The above definitions are sufficient to determine approximately the values of the sine and cosine. Thus it is possible to produce a graph of these functions. Here is the graph of the function y = sin(x) on the interval
Now here is the graph of the function y = cos
As for the other functions, one can obtain graphs for them also. The function x → tan
Finally, the graph of x → sec
Both of these functions have vertical asymptotes at odd multiples of π∕2 although I have not shown them with the secant function.
The formula for the cosine and sine of the sum of two angles is also important. Like most of this material, I assume the reader has seen it. However, I am aware that many people do not see these extremely important formulas, or if they do, they often see no explanation for them so I shall give a review of it here.
The following theorem is the fundamental identity from which all the major trig. identities involving sums and differences of angles are derived.
Proof: Recall that for a real number z, there is a unique point p
Also from basic geometric reasoning, the distance between the points p

Expanding the above,


Now using that cos^{2} + sin^{2} = 1,

Therefore,
