The Art of Interface

Article 11 — Appendix A.28

sin — trigonometric sine function

Category. Mathematics.

Abstract. Trigonometric sine: definition, plot, properties, identities and table of values for some angles.

Reference. This article is a part of Librow scientific formula calculator project.

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1. Definition

Sine of the angle is ratio of the opposite leg to hypotenuse.

2. Plot

Sine is 2π periodic function defined everywhere on real axis — so its wave-like plot spreads endlessly to the left and to the right.

Fig. 1. Plot of the sine function y = sin x. Fig. 1. Plot of the sine function y = sinx.

Function codomain is limited to the range [−1, 1].

3. Identities

Base:

sin2φ + cos2φ = 1

and its consequences:

sinφ = ±√(1 − cos2φ)
sinφ = ±tanφ /√(1 + tan2φ)
sinφ = ±1 /√(1 + cot2φ)
sinφ = ±√(sec2φ − 1) /secφ

By definition:

sinφ ≡ 1 /cscφ

Properties — symmetry, periodicity, etc.:

sin−φ = −sinφ
sinφ = sin(φ + 2πn), where n = 0, ±1, ±2, ...
sinφ = sin(π − φ)
sinφ = −sin(π + φ)
sinφ = cos(π/2 − φ)

Half-angle:

sin(φ/2) = ±√[(1 − cosφ) /2]
sinφ = 2 tan(φ/2) /[1 + tan2(φ/2)]

Double angle:

sin(2φ) = 2 sinφ cosφ
sin(2φ) = 2 tanφ /(1 + tan2φ)

Triple angle:

sin(3φ) = 3 cos2φ sinφ − sin3φ = 3 sinφ − 4 sin3φ

Quadruple angle:

sin(4φ) = cosφ (4 sinφ − 8 sin3φ)

Power reduction:

sin2φ = [1 − cos(2φ)] /2
sin3φ = [3 sinφ − sin(3φ)] /4
sin4φ = [3 − 4 cos(2φ) + cos(4φ)] /8
sin5φ = [10 sinφ − 5 sin(3φ) + sin(5φ)] /16
sin2φ cos2φ = [1 − cos(4φ)] /8
sin3φ cos3φ = [3 sin(2φ) − sin(6φ)] /32
sin4φ cos4φ = [3 − 4 cos(4φ) + cos(8φ)] /128
sin5φ cos5φ = [10 sin(2φ) − 5 sin(6φ) + sin(10φ)] /512

Sum and difference of angles:

sin(φ + ψ) = sinφ cosψ + cosφ sinψ
sin(φ − ψ) = sinφ cosψ − cosφ sinψ

Product-to-sum:

sinφ sinψ = [cos(φ − ψ) − cos(φ + ψ)] /2
sinφ cosψ = [sin(φ + ψ) + sin(φ − ψ)] /2

Sum-to-product:

sinφ + sinψ = 2 sin[(φ + ψ) /2] cos[(φ − ψ) /2]
sinφ − sinψ = 2 sin[(φ − ψ) /2] cos[(φ + ψ) /2]
sinφ + sin(φ + ψ) + sin(φ + 2ψ) + ... + sin(φ + nψ) = sin[(n + 1) ψ/2] sin(φ + nψ/2) /sin(ψ/2)

Sine of inverse functions:

sin(arcsin x) ≡ x
sin(arccos x) = √(1 − x2)
sin(arctan x) = x /√(1 + x2)

Some angles:

Angle φValue sinφ
00
π/12(√6 − √2) /4
π/10(√5 − 1) /4
π/8√(2 − √2) /2
π/61 /2
π/5√(10 - 2√5) /4
π/41 /√2
3π/10(√5 + 1) /4
π/3√3 /2
3π/8√(2 + √2) /2
2π/5√(10 + 2√5) /4
5π/12(√6 + √2) /4
π/21
Table 1. Sine for some angles.

4. Support

Trigonometric sine function sin of the real argument is supported by free version of the Librow calculator.

Trigonometric sine function sin of the complex argument is supported by professional version of the Librow calculator.

5. How to use

To calculate sine of the number:

sin(-1);

To calculate sine of the current result:

sin(rslt);

To calculate sine of the angle φ in memory:

sin(mem[φ]);