Trigonometry & The Unit Circle
trig is understood until you can do it forwards and backwards.
In the earlier discussion of right triangle trigonometry in Euclidean Geometry without or before coordinates, we saw how similarity implied the computation of sine and cosine for acute angles only depended on the angle and not on the scale or size of the right triangle drawn for the computation. All such triangles were similar.
The next page sections first show how to compute sine and cosine for acute angles with the aid of a unit circle. Then they observe the computations work for angles that are not acute as well. This leads to a unit circle computation of sine and cosines for angles, acute or not, which is consistent with (agrees with) the right triangle approach for acute angles.
So the unit circle method extends the domain on which sine and cosine can be computed. Or, in retrospect, we may pivot or turn around and say that unit circle approach when restricted to acute angles yields the right triangle approach as a special case.
Students who go beyond calculus (the subject of slope related computations) may see a further pivot or change of perspective in sine and cosine (and exponentials) are computed as limits of sequences of polynomial approximations, best written when angles are measured in terms of radians, and in which angles may have complex values. The unit circle viewpoint in a step in that direction.
(I) Extension of SINE and Cosine
with help of a unit circle
A Forward Direction. From special case of acute angles to general case where angles do not have to be acute.
For an acute angle q , any right triangle with angle q may be used to compute sine and cosine for that angle q. In steps 1 to 3 we choose a right triangle drawn in the first quadrant of a coordinate plane. That leads to a triangle-free way to compute sine and cosine, a way that agrees with the right triangle computation method but which also works for angles that are not acute.
Step 1.
Draw a unit circle
Your unit of measurement may be one centimeter, one meter, one kilometer, one inch, one foot, one yard, one mile or any other unit. Choose one, or draw a circle and declare its radius to be your unit length.
Step 2.
Let q be an acute angle. Locate the head of the vector with angle q and length 1 on the unit circle.
Step 3.
The head will have coordinates (a units, b units)
Now right triangle trigonometry gives cos(q) = a and sin (q) = b and hence
Further trig functions may be computed as follows.
An Alternate Way to Compute cosine and sine
Therefore locating the point with angle q on the unit circle.
and declaring cos(q) = a and sin (q) = b gives an alternate way to compute the cosine and sine of an acute angle q, which is consistent with our earlier right triangle definition when ngle q is acute, but this method also works when angle q is not between 0 and 90 degrees. For example if the angle is between 90 and 180 degrees, the point (a,b) will be in quadrant II
and we take cos(q) = a and sin (q) = b. Here the connection with the sides of an acute angle right triangle is lost.
Remark 1. (The Pythagorean Identity): From a2 + b2 = 1, we get [cos(q)]2 + [sin(q)]2 = 1 or
Remark 2. (Period of Sine and Cosine): Adding 360 degrees to an angle q that determines a point (a,b) on the unit circle give another angle which determines the same point. So sine and cosine have period 360 degrees.
Remark 3. (Effect of Similarity). If we change the unit length, our unit circle will change. But similarity of triangles and circles implies the values of the coordinates a and b will not change. How and why is left as an exercise.
II. Step Reversal.
Direction Reversal. From general case where angles do not have to be acute to the special case of acute angles to
A. Trigonometry
The simplest way to introduce trigonometric functions (functions on your calculator) is to begin with their unit circle definitions, and then specialize to their right triangle computation with the help of similarity assumptions about triangles, right or scalene. Several steps follow for reading in or besides your trig course.
Step 1.
Draw a unit circle
Your unit of measurement may be one centimeter, one meter, one kilometer, one inch, one foot, one yard, one mile or any other unit. Choose one, or draw a circle and declare its radius to be your unit length.
Exercise for Later: How does similarity assumptions for right triangles imply the results, here the definition of trig functions below, is independent of the choose of unit length?
Step 2.
Let q be an angle. Locate the head of the vector with angle q and length 1 on the unit circle.
Step 3.
The head will have coordinates (a units, b units)
on circle of radius 1 unit.
Put cos(q) =a and sin (q) =b. This defines both sine and cosine for all values of the angle q.
Further trig functions may be defined as follows.
when the divisors are nonzero.
The case where q is between 0 and 90 degrees is considered next.
Step 4 (Right Triangle Trigonometry)
circle of radius 1
unit.
Assume q is between 0 and 90 degrees. Then
For angles between 0 and 90 degrees, similarity of right triangles implies the ratios
if you replace the unit circle right triangle by a similar right triangle.
The latter formulas for may be used to compute 
with any right triangle where sides are labeled opposite and adjacent for an
angle 
The further
trig functions may be defined as follows.
when the divisors are nonzero.
Exercise: Express these further trig functions as ratios of the sides opposite, adjacent and/or hypotenuse of the above right triangle.
A trig course will explain the following in more detail.
Trig functions link the ratio of two sides of a right triangle to cosines, sines and tangents of an angle. Knowledge of two sides in right triangle gives knowledge of the third by means of Pythagorean theorem, and of the values of the trig functions for the angles in the triangle. Computation of unknown side lengths, unknown hypotenuse lengths and unknown angles is useful in land measurement (geo - metry) and also in navigation.
From one-to-one properties of trig functions for angles between 0 and 90 degrees or ½p, one can define (say how to compute) inverse trig functions (more functions on your calculator) to compute the angles from the ratio of sides. Computation with inverse trig functions allows one to obtain polar coordinates for vectors or complex numbers from coordinates, real and imaginary parts, or the length of the adjacent and opposite sides of a right triangle determined by the coordinates. Again, this removes the need to measure the lengths and angles for points with rectangular coordinates [a, b].
Calculation
One may define trig functions by saying how to compute them in principle as above, but then one computes or approximates them in practice from tables and slide rules (old fashioned approach) or using calculators (the new approach). Unfortunately in this practice, the tables, slide rules or calculation devices are black boxes which provide results, but whose derivation or justification is not commonly known. This departs from the principle of understanding the computations one does, but the numbers computed by these black boxes can be checked in simple cases. When calculators first arrived, some used faulty or suboptimal methods (algorithms) to compute. Beware.