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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, formula relating numerator terms, 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, formula relating denominator terms for the coefficients of the terms of the denominator. See also:

• De Moivre's Theorem
• Chebyshev's Method
• Multiple Angles Cosines
• Multiple Angles Sines
• Multiple Angles Tangents
• De Moivre's Theorem Extended

Trigonometry Contents

Page Contents

1. List of Formulae
2. Table of Tangents
3. Observations from the Table
4. Proof

List of Formulae

tan6x
tan7x
tan8x
tan9x
tan10x

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
		k	0	1	2	3	4	5	Formula
n	0	Top	0	.					tan(0)=0
		Bottom	1
	1	Top	1						tan(1x)=tan(x)
		Bottom	1
	2	Top	2
		Bottom	1	-1
	3	Top	3	-1
		Bottom	1	-3
	4	Top	4	-4
		Bottom	1	-6	1
	5	Top	5	-10	1
		Bottom	1	-10	5
	6	Top	6	-20	6				tan6x
		Bottom	1	-15	15	-1
	7	Top	7	-35	21	-1			tan7x
		Bottom	1	-21	35	-7
	8	Top	8	-56	56	-8			tan8x
		Bottom	1	-28	70	-28	1
	9	Top	9	-84	126	-36	1		tan9x
		Bottom	1	-36	126	-84	9
	10	Top	10	-120	252	-120	10		tan10x
		Bottom	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:

the coefficient is 5
2. When n is even, the numerator coefficients and the denominator coefficients are symmetrical: For instance, when n=10: tan10x
   We notice the numerator coefficients are 10, -120, 252, -120,10, and the denominator coefficients are: 1, -45, 210, -210, 45, -1
3. The first coefficient in the denominator is always 1.
4. The signs alternate.
5. When n is odd, the last coefficient in the numerator is ±1. When n is even it is ±n
6. When n is even, the last coefficient in the denominator is ±1. When n is odd it is ±n.
7. The relationship between the coefficient of a term, n, k and the coefficient of term n-1, k is
   tanTermsTop.gif
   Where tanHnk.gif is the coefficient of the kth term of the nth numerator, and tanKnk.gif is the coefficient of the kth term of the nth denominator. And
   tantermsBottom.gif
   
   

Proof

Trigonometry Contents

Ken Ward's Mathematics Pages

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