The Art of Mathematics

Help 6.21

cos — trigonometric cosine function

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 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(φ − ψ) = cosφ cosψ + sinφ sinψ

Product-to-sum:

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

Sum-to-product:

cosφ + cosψ = 2 cos[(φ + ψ) /2] cos[(φ − ψ) /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. Derivative and indefinite integral

Cosine derivative:

cos′x = −sinx

Indefinite integral of the cosine:

∫ cosx dx = sinx + C

where C is an arbitrary constant.

5. How to use

To calculate cosine of the number:

cos(−1);

To get cosine of the complex number:

cos(−1+i);

To get cosine of the current result:

cos(rslt);

To get cosine of the angle φ in calculator memory:

cos(mem[φ]);

6. Support

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

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