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.
Sine of the angle is ratio of the opposite leg to hypotenuse.
Sine is 2π periodic function defined everywhere on real axis — so its wave-like plot spreads endlessly to the left and to the right.
[画像: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].
Base:
sin2φ + cos2φ = 1and its consequences:
sinφ = ±√(1 − cos2φ)By definition:
sinφ ≡ 1 /cscφProperties — symmetry, periodicity, etc.:
sin−φ = −sinφHalf-angle:
sin(φ/2) = ±√[(1 − cosφ) /2]Double angle:
sin(2φ) = 2 sinφ cosφTriple angle:
sin(3φ) = 3 cos2φ sinφ − sin3φ = 3 sinφ − 4 sin3φQuadruple angle:
sin(4φ) = cosφ (4 sinφ − 8 sin3φ)Power reduction:
sin2φ = [1 − cos(2φ)] /2Sum and difference of angles:
sin(φ + ψ) = sinφ cosψ + cosφ sinψProduct-to-sum:
sinφ sinψ = [cos(φ − ψ) − cos(φ + ψ)] /2Sum-to-product:
sinφ + sinψ = 2 sin[(φ + ψ) /2] cos[(φ − ψ) /2]Sine of inverse functions:
sin(arcsin x) ≡ xSome angles:
| Angle φ | Value sinφ |
|---|---|
| 0 | 0 |
| π/12 | (√6 − √2) /4 |
| π/10 | (√5 − 1) /4 |
| π/8 | √(2 − √2) /2 |
| π/6 | 1 /2 |
| π/5 | √(10 - 2√5) /4 |
| π/4 | 1 /√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/2 | 1 |
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.
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]);