# Values of Trigonometric Functions

While the integers, rational numbers, and algebraic numbers can stand on their own, for the purpose of this discussion, we will consider them -- together with the transcendental numbers -- as subsets of the real numbers. From this point of view, rational numbers are those which can be represented as a quotient of two, relatively prime, integers. The algebraic numbers are those which are the roots of monic polynomials with rational coefficients. Alternatively, of polynomials with integral coefficients. Finally, the transcendental numbers are those real numbers, which are not algebraic. Furthermore, these sets of numbers are nested: Integers are rational numbers and rational numbers are algebraic numbers. We define the irrational numbers as the complement of the rational numbers in the real numbers.

The rational numbers, algebraic numbers, and transcendental numbers are closed under rational operations.

A transcendental function is such a function on the Real into Real which maps the rational numbers into the transcendental numbers. It can be shown that the exponential function and its inverse -- the logarithmic function -- are transcendental functions. It follows that pi (which is defined as the smallest positive number such that exp(2 pi i) = 1) and epsilon (which is defined as epsilon = exp(1)) are transcendental.

Since the trigonometric functions are rational functions of the exponential function, they also are transcendental functions. However, we show that the circular trigonometric functions of rational multiples of pi are algebraic numbers, as long as the denominator is either a non-negative power of two or three times a non-negative power of two and the numerator is an integer..

Trigonometric Function of Multiples of pi / 24
x | sin(x) | cos(x) | tan(x) | hav(x)
--- | --- | --- | --- | ---
- 2 pi | 0 | 1 | 0 | 0
- 7 pi / 4 | sqrt(2) / 2 | sqrt(2) / 2 | 1 | (2 - sqrt(2)) / 2
- 3 pi /2 | 1 | 0 | oo | 1 / 2
- 5 pi / 4 | sqrt(2) / 2 | - sqrt(2) / 2 | - 1 | (2 + sqrt(2)) / 2
- pi | 0 | -1 | 0 | 1
- 3 pi / 4 | - sqrt(2) / 2 | - sqrt(2) / 2 | 1 | (2 + sqrt(2)) / 2
- pi / 2 | -1 | 0 | oo | 1 / 2
- pi / 4 | - sqrt(2) / 2 | sqrt(2) / 2 | - 1 | (2 - sqrt(2)) / 2
0 | 0 | 1 | 0 | 0
pi / 24 | sqrt(8 - 2 sqrt(2) - 2 sqrt(6)) / 4 | sqrt(8 + 2 sqrt(2) + 2 sqrt(6)) / 4 |  |
pi / 12 | sqrt(2 - sqrt(3)) / 2 | sqrt(2 + sqrt(3)) / 2 | 2 - sqrt(3) | (2 - sqrt(2 + sqrt(3))) / 4
pi / 8 | sqrt(2 - sqrt(2)) / 2 | sqrt(2 + sqrt(2)) / 2 | (2 - sqrt(2)) / 2 | (2 - sqrt(2 + sqrt(2))) / 4
pi / 6 | 1 / 2 | sqrt(3) / 2 | sqrt(3) / 3 | (2 - sqrt(3)) / 4
5 pi / 24 | sqrt(8 + 2 sqrt(2) - 2 sqrt(6)) / 4 | sqrt(8 - 2 sqrt(2) + 2 sqrt(6)) / 4 |  |
pi / 4 | sqrt(2) / 2 | sqrt(2) / 2 | 1 | (2 - sqrt(2)) / 2
7 pi / 24 | sqrt(8 - 2 sqrt(2) + 2 sqrt(6)) / 4 | sqrt(8 + 2 sqrt(2) - 2 sqrt(6)) / 4 |  |
pi / 3 | sqrt(3) / 2 | 1 / 2 | sqrt(3) | 1 / 4
3 pi / 8 | sqrt(2 + sqrt(2)) / 2 | sqrt(2 - sqrt(2)) / 2 | (2 + sqrt(2)) / 2 | (2 - sqrt(2 - sqrt(2))) / 4
5 pi / 12 | sqrt(2 + sqrt(3)) / 2 | sqrt(2 - sqrt(3)) / 2 | 2 + sqrt(3) | (2 - sqrt(2 - sqrt(3))) / 4
11 pi / 24 | sqrt(8 + 2 sqrt(2) + 2 sqrt(6)) / 4 | sqrt(8 - 2 sqrt(2) - 2 sqrt(6)) / 4 |  |
pi / 2 | 1 | 0 | oo | 1 / 2
3 pi / 4 | sqrt(2) / 2 | - sqrt(2) / 2 | - 1 | (2 + sqrt(2)) / 2
pi | 0 | -1 | 0 | 1
5 pi / 4 | - sqrt(2) / 2 | - sqrt(2) / 2 | 1 | (2 + sqrt(2)) / 2
3 pi / 2 | -1 | 0 | oo | 1 / 2
7 pi / 4 | - sqrt(2) / 2 | sqrt(2) / 2 | - 1 | (2 - sqrt(2)) / 2
2 pi | 0 | 1 | 0 | 0

The circular trigonometric function of any -- subject to the aforementioned restriction -- rational multiple of pi may be found algebraically.

Proof of the values in the foregoing table: Obtain the tangent function by dividing the sine by the cosine and rationalizing the denominator. Obtain the haversine function from its definition as (1 - cos(x)) / 2.

The sine and cosine functions may be obtained as follows. At x = 0, employ the McLaurin series. The sine and cosine functions have a modulus of periodicity of 2 pi; hence, these functions have the same respective values at +- 2 pi as at zero. At pi, then at pi / 2, employ the half-angle formulae and resolve the signs by employing the McLaurin's series.

Two angles are said to be supplementary iff (= if and only if) their sum is pi. Let the angles x and y be supplementary. Solve for y = pi - x. Employ the addition theorem to prove that for y supplementary to x, sin(y) = sin(x), cos(y) = - cos(x), and tan(y) = - tan(x). These relationships propagate the values of the trigonometric functions out from the first quadrant.

Two angles are said to be complementary iff (=if and only if) their sum is pi / 2. Let the angles x and y be complementary. Solve for y = pi / 2 - x. Employ the addition theorem to prove that for y complementary to x, sin(y) = cos(x) and tan(y) = 1 / tan(x). These relationships propagate the values from the interval [0, pi / 4] onto the interval [pi / 4, pi / 2]. Also by employing the Pythagorean theorem and the Law of Sines, it shows that, in a plane, for a right triangle, with a hypotenuse of one, the leg opposite to the acute angle x has a magnitude of sin(x), while the leg adjacent has a magnitude of cos(x).

From this diagram, one may draw a crude graph of the trigonometric functions. The graph suffices to locate the zeros and poles of the trigonometric functions. Also, their signs are apparent.

Consider an isosceles right triangle, with the magnitude of each leg equal to one. Then the magnitude of the hypotenuse is sqrt(2). From this triangle, the values of the trigonometric functions of pi / 4 follow.

Consider an equilateral triangle, with the magnitude of each side equal to two. Drop a perpendicular from the apex to the base. Each of the small triangles is a right triangle, with magnitudes of the legs equal to 1 or sqrt(3) / 2. From this triangle, the values of the trigonometric function of pi / 6 follow.

Employ the half-angle formulae to obtain the values of the trigonometric functions at pi / 8 and at pi / 12.

The obvious way of obtaining the values of the trigonometric functions at pi / 24 is by employing the half-angle formulae. Try it -- it is an instructive exercise. But the easy way is to write

pi / 24 = pi / 6 - pi / 8

and employ the addition theorems. Then square the result, simplify it, and take its square root.

Likewise, write

7 pi / 24 = pi / 8 + pi / 6

to obtain the values of the trigonometric functions of 7 pi / 24. Or write

5 pi / 24 = 3 pi / 8 - pi / 6

to obtain the values of the trigonometric function of 5 pi / 24. In either case, employ the relationship of complementary angles to obtain the other. There are other possibilities; but these are the easiest.

The remaining entries may be obtained by employing the addition theorems for the sine and cosine functions.

We have omitted those entries for the tangent, for which rationalization of the denominator only complicates the expression.

The values in this table may be checked by observing that, in each row, the sum of the squares of the sine and cosine must be one. Also, employ the double-angle formulae to check each entry. The only entries that cannot be checked by the double angle formulae are those for pi / 3 -- whose values you have to know -- and for 2 pi. This last row has to be the same as the row for zero -- whose values you should know.

It would be easy enough to expand the foregoing table to list each of the 97 multiples of pi / 24 in the domain [- 2 pi, 2 pi].

Decreasing the step-size below pi / 24 would be difficult; because each time we employ the half-angle formula, we add another level of square-root. As you have observed, going from pi / 12 to pi / 24 is very difficult -- that is why we went from pi / 6 - pi / 8, instead. But, once we arrive at pi / 24, the domain is closed under addition and subtraction; thus, we cannot employ the addition theorems.

In our trigonometry class, we were required to know the derivation of each of these values in the foregoing table. We were encouraged to memorize most of the values.

How did we decide what to include in the table?
We only provided representative values outside of the first quadrant.
In the first quadrant, we wanted to provide *all* multiples of some primitive angle.
We wanted to include the multiples of pi/12.
Multiples of any of these angles -- pi/2, pi/3, pi/4, pi/6, pi/8, pi/12, or pi/16 -- would not display all of the techniques.
We are left with the multiples of pi/24, which is what we employed.
Multiples of pi/48, or higher, would not involve any new techniques.

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Copyright © 1999, 2000, 2 R. I. 'Scibor-Marchocki last modified on Thursday 07-th November 2002.