  # Ken Ward's Mathematics Pages

## Trigonometry Tangent Multiple Angle Formulae

This page lists the formulae for tan nx for n=2, up to n=10. By studying the table of coefficients of these polynomials, we can infer other formulae. For instance, , relating the coefficients of the terms in the numerator (H) of the tan nx formula (or numbers in the table) to the numerator and denominator (K) of tan (n-1)x, thus enabling us to compute the coefficients of tan nx from those of tan(n-1)x. Similarly, for the coefficients of the terms of the denominator. See also:
Trigonometry Contents

## List of Formulae         ## Table of Tangents

The formula repeated from above are meant to clarify the table. The powers are written, for instance, 1/0, where the top figure relates to the power of the corresponding term in the numerator (top), and 0 relates to the power of the corresponding coefficient of the term in the denominator (bottom). The term number, k, is 0, 1, 2...
Power 1/0 (3/2 5/4 7/6 9/8 11/10 Formula 0 .  tan(0)=0 1 1 tan(1x)=tan(x) 1 2 1 -1 3 -1 1 -3 4 -4 1 -6 1 5 -10 1 1 -10 5 6 -20 6 1 -15 15 -1 7 -35 21 -1 1 -21 35 -7 8 -56 56 -8 1 -28 70 -28 1 9 -84 126 -36 1 1 -36 126 -84 9 10 -120 252 -120 10 1 -45 210 -210 45 -1

## Observations from the Table

What follows are observations, not all of which are proved on this page. At this stage, they are hypotheses (if we are thinking scientically about the data in the table) or conjectures (if we are thinking mathematically). [They are all provable, however]

There are many patterns in the above table and formulae, some of which are mentioned below.
1. The first coefficient of the numerator (coefficient of tan x) is always n. For instance, when n=5:
2. the coefficient is 5
3. When n is even, the numerator coefficients and the denominator coefficients are symmetrical: For instance, when n=10: We notice the numerator coefficients are 10, -120, 252, -120,10, and the denominator coefficients are: 1, -45, 210, -210, 45, -1
4. The first coefficient in the denominator is always 1.
5. The signs alternate.
6. When n is odd, the last coefficient in the numerator is ±1. When n is even it is ±n
7. When n is even, the last coefficient in the denominator is ±1. When n is odd it is ±n.
8. The relationship between the coefficient of a term, n, k and the coefficient of term n-1, k is Where is the coefficient of the kth term of the nth numerator, and is the coefficient of the kth term of the nth denominator. And ## Proof

Trigonometry Contents

Ken Ward's Mathematics Pages

# Faster Arithmetic - by Ken Ward

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: 