Appendix 3

Some Trigonometric Gems

“Beauty is in the eye of the beholder,” says an old proverb. I have collected here a sample of trigonometric formulas that will appeal to anyone’s sense of beauty. Some of these formulas are easy to prove, others will require some effort on the reader’s behalf. My selection is entirely subjective: trigonometry abounds in beautiful formulas, and no doubt the reader can find many others that are equally appealing.

1. FINITE FORMULAS

sin² α + cos² α = 1

sin⁴ α − cos⁴ α = sin² α − cos² α

sec² α + csc² α = sec² α csc² α

sin (α + β) sin (α − β) = sin² α − sin² β

tan (45^° + α) tan (45^° − α) = cot (45^° + α) cot (45^° − α) = 1

sin (α + β + γ) + sin α sin β sin γ = sin α cos β cos γ + sin β cos γ cos α + sin γ cos α cos β

Let f(α, β) = cos² α + sin² α cos 2β;

then f(α, β) = f(β, α).

Let g(α, β) = sin² α − cos² α cos 2β;

then g(α, β) = g(β, α).

In the following relations, let α + β + γ = 180^°:

sin α + sin β + sin γ = 4 cos α/2 cos β/2 cos γ/2

sin 2α + sin 2β + sin 2γ = 4 sin α sin β sin γ

sin 3α + sin 3β + sin 3γ = −4 cos 3α/2 cos 3β/2 cos 3γ/2

cos α + cos β + cos γ = 1 + 4 sin α/2 sin β/2 sin γ/2

cos² 2α + cos² 2β + cos² 2γ − 2 cos 2α cos 2β cos 2γ = 1

tan α + tan β + tan γ = tan α tan β tan γ¹

with equality if, and only if, α = β = γ = 60^°.

In any acute triangle,

with equality if, and only if, α = β = γ = 60^°.

In any obtuse triangle,

−∞ < tan α + tan β + tan γ < 0.

2. INFINITE FORMULAS²

sin x = x − x³/3! + x⁵/5! − + · · ·

cos x = 1 − x²/2! + x⁴/4! − + · · ·

sin x = x(1 − x²/π²)(1 − x²/4π²)(1 − x²/9π²) · · ·

cos x = (1 − 4x²/π²)(1 − 4x²/9π²)(1 − 4x²/25π²) · · ·

tan x = 8x[1/(π² − 4x²) + 1/(9π² − 4x²) + 1/(25π² − 4x²) + · · ·]

sec x = 4π[1/(π² − 4x²) − 3/(9π² − 4x²) + 5/(25π² − 4x²) − + · · ·]

(sin x)/x = cos x/2 cos x/4 cos x/8 · · ·

(1/4) tan π/4 + (1/8) tan π/8 + (1/16) tan π/16 + · · · = 1/π

tan⁻¹ x = x − x³/3 + x⁵/5 − + · · ·,    −1 < x < 1.

Notes

1. The companion formula

cot α + cot β + cot γ = cot α cot β cot γ

holds true only for α + β + γ = 90^°.

2. For a sample of Fourier series, see figure 95, p. 206. Numerous other trigonometric series can be found in Summation of Series, collected by L. B. W. Jolley (1925; rpt. New York: Dover, 1961), chaps. 14 and 16.