Maths - Trigonometry

Trigonometry is about angles and triangles, there are relationships between the angles and the ratios of the lengths of side of the triangles. These ratios are represented by functions such as sine and cosine, which occur widely in mathematics and physics, even in fields which don't initially appear to be related to physical triangles.

On these pages we will consider two types of trigonometry:

• trigonometry based on a circle where: a²=b²+c²
• trigonometry based on a hyperbola where: a²=b²-c²

For most cases we only need to consider traditional trigonometry based on a circle so feel free to ignore the hyperbolic trigonometry column. Hyperbolic trigonometry starts to become useful when we have a space with the Minkowski norm: (x²-y²) the simplest case is the two dimensional space represented by double numbers as explained here.

Summary of results

This is a summary of the results in the above pages, for the derivation of these formulae see the links above.

Right angled triangle

circle based trigonometry	hyperbolic trigonometry
Here we will use a unit circle, that is a circle where all the points are exactly 1 unit from the origin, which gives: b²+c²=1.	Here we will use a hyperbolic function where the minimum distance from the origin is 1 unit which gives: b²-c²=±1.
sine function sin(θ) = b	hyperbolic sine function sinh(θ) = b
cosine function cos(θ) = c	hyperbolic cosine function cosh(θ) = c
tangent function tan(θ) = b / c	hyperbolic tangent function tanh(θ) = d = b/c
cosecant cosec(θ) = a / b
secant sec(θ) = a / c
cotangent cot(θ) = c / b

Trig Identities

The values of sin, cos and tan are related to each other, for instance,

hyperbolic

sinh(t) = (e^t - e^-t )/2 = (e^t - 1)/2*e^t = -i sin(i t)
cosh(t) = (e^t + e^-t )/2 = (e^t + 1)/2*e^t = cos(i t)
tanh(t) = (e^t - e^-t )/(e^t + e^-t ) = (e^2t - 1)/(e^2t + 1)= -i tan(i t)
asinh(x) = ln(x+√(x²+1))
acosh(x) = if (abs(x)<1) ln(x+√(1 - x²))
else ln(x+√(x²-1))
atanh(x) = if (abs(x)<1) 0.5*ln((x-1)/(x+1))
else 0.5*ln((x-1)/(x+1))

Complimentary and Supplementary Angles

• sin(θ - 90°) = cos(θ)
• cos(θ - 90°) = sin(θ)
• tan(θ - 90°) = cot(θ)

Non-Right Angled Triangles - The Sine Rule

a / sin A = b / sin B = c / sin C

Non-Right Angled Triangles - The Cosine Rule

• cos A = (b² + c² - a²)/2*b*c
• cos B = (c² + a ² - b ²)/2*c*a
• cos C = (a² + b ² - c ²)/2*a*b

Area of triangles

In euclidean space the area of a triangle depends on length of 2 sides and one angle:

area = ¹/₂ b * c*sin A

This is half the product of two sides and the sin of the included angle.

In hyperbolic space the area depends only on the internal angles:

π-(α+β+γ)=CΔ

Addition and Subtraction formulae

Products of Trigonometric Functions

Double Angle and Half Angle Formulae

We can derive the double angle from products above where A=B. From this we can then derive the half angle formulae.

derivation on this page.

Angles of Any Magnitude

• sin(0) = 0
• cos(0) = 1
• cos(90° + θ) = -sin(θ)
• cos(90° - θ) = sin(θ)
• sin(180° - θ) = sin(θ)
• sin(180° + θ) = -sin(θ)
• sec(180° + θ) = -sec(θ)
• cosec(180° - θ) = cosec(θ)
• cos(270° + θ) = sin(θ)
• cos(270° - θ) =
• tan(270° + θ) = -cotan(θ)
• tan(270° - θ) = cotan(θ)
• cot(360° + θ) = cot(θ)
• sin(360° - θ) = -sin(θ)

Series

• sin(a) = a - ( a ³ / 3!) + ( a ⁵ / 5!) - ( a ⁷ / 7!) + ...
• cos(a) = 1 - ( a ² / 2!) + ( a ⁴ / 4!) - ( a ⁶ / 6!) + ...
• tan(a) = a + ( a ³ / 3) + ( 2a⁵ / 15) + ...
• sin^-1(x) = x + ( x³ / 3!) + ( 9x⁵ / 5!) + (225 x⁷ / 7!) + ...
• cos^-1(x) = PI/2 - x - ( x³ / 3!) - (9 x⁵ / 5!) - ( 225 x⁷ / 7!) + ...
• tan^-1(x) = x - ( x³ / 3) + ( x⁵ / 5) - ( x⁷ / 7) + ...

Other topics are as follows:

As a starting point goto this page: triangles

Prerequisites

On this page we discussed how to represent and evaluate functions.

Hyperbola

Example Dish Example