The Helpful Mathematics
+38 044 572 93 47

Article 11 — Appendix A.28

sin — trigonometric sine function

Category. Mathematics.

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

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

1. Definition

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

2. Graph

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

Fig. 1. Graph of the sine function y = sin x Fig. 1. Graph 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(φψ) = sinφ cosψ − cosφ sinψ

Product-to-sum:

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

Sum-to-product:

sinφ + sinψ = 2 sin[(φ + ψ) /2] cos[(φψ) /2]
sinφ − sinψ = 2 sin[(φψ) /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 is supported in:

Trigonometric sine function of the complex argument sin is supported in:

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]);