The Art of Mathematics

Help 6.42

sin — trigonometric sine function

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 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(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. Derivative and indefinite integral

Sine derivative:

sin′x = cosx

Indefinite integral of the sine:

∫ sinx dx = −cosx + C

where C is an arbitrary constant.

5. How to use

To calculate sine of the number:

sin(−1);

To get sine of the complex number:

sin(−1+i);

To get sine of the current result:

sin(rslt);

To get sine of the angle φ in calculator memory:

sin(mem[φ]);

6. Support

Trigonometric sine of the real argument is supported in free version of the Librow calculator.

Trigonometric sine of the complex argument is supported in professional version of the Librow calculator.