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Article 11 — Appendix A.15

cos — trigonometric cosine function

Category. Mathematics.

Abstract. Trigonometric cosine: 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

Cosine of the angle is ratio of the adjacent leg to hypotenuse.

2. Graph

Cosine 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 cosine function y = cos x Fig. 1. Graph of the cosine function y = cosx.

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

3. Identities

Base:

sin2φ + cos2φ = 1

and its consequences:

cosφ = ±√(1 − sin2φ)
cosφ = ±1 /√(1 + tan2φ)
cosφ = ±cotφ /√(1 + cot2φ)
cosφ = ±√(csc2φ − 1) /cscφ

By definition:

cosφ ≡ 1 /secφ

Properties — symmetry, periodicity, etc.:

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

Half-angle:

cos(φ/2) = ±√[(1 + cosφ) /2]
cosφ = [1 − tan2(φ/2)] /[1 + tan2(φ/2)]

Double angle:

cos(2φ) = cos2φ − sin2φ
cos(2φ) = 2 cos2φ − 1
cos(2φ) = 1 − 2 sin2φ
sin(2φ) = (1 − tan2φ) /(1 + tan2φ)

Triple angle:

cos(3φ) = cos3φ − 3 sin2φ cosφ = 4 cos3φ − 3 cosφ

Quadruple angle:

cos(4φ) = 8 cos4φ − 8 cos2φ + 1

Power reduction:

cos2φ = [1 + cos(2φ)] /2
cos3φ = [3 cosφ + cos(3φ)] /4
cos4φ = [3 + 4 cos(2φ) + cos(4φ)] /8
cos5φ = [10 cosφ + 5 cos(3φ) + cos(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:

cos(φ + ψ) = cosφ cosψ − sinφ sinψ
cos(φψ) = cosφ cosψ + sinφ sinψ

Product-to-sum:

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

Sum-to-product:

cosφ + cosψ = 2 cos[(φ + ψ) /2] cos[(φψ) /2]
cosφ − cosψ = −2 sin[(φ + ψ) /2] sin[(φ - ψ) /2]
cosφ + cos(φ + ψ) + cos(φ + 2ψ) + ... + cos(φ + nψ) = sin[(n + 1) ψ/2] cos(φ + nψ/2) /sin(ψ/2)

Cosine of inverse functions:

cos(arccos x) ≡ x
cos(arcsinx) = √(1 − x2)
cos(arctan x) = 1 /√(1 + x2)

Some angles:

Angle φValue cosφ
01
π/12(√6 + √2) /4
π/10√(10 + 2√5) /4
π/8√(2 + √2) /2
π/6√3 /2
π/5(√5 + 1) /4
π/41 /√2
3π/10√(10 - 2√5) /4
π/31 /2
3π/8√(2 − √2) /2
2π/5(√5 − 1) /4
5π/12(√6 − √2) /4
π/20
Table 1. Cosine for some angles.

4. Support

Trigonometric cosine function cos is supported in:

Trigonometric cosine function of the complex argument cos is supported in:

5. How to use

To calculate cosine of the number:

cos(-1);

To calculate cosine of the current result:

cos(Rslt);

To calculate cosine of the angle φ in memory:

cos(Mem[φ]);