Spherical trigonometry

A great-circle arc, on the sphere, is the analogue of a straight line, on the plane.

Where two such arcs intersect, we can define the
**spherical angle** *either* as angle between the
tangents to the two arcs, at the point of intersection, *or*
as the angle between the planes of the two great circles where they
intersect at the centre of the sphere.

(Spherical angle is only
defined where arcs of *great *circles meet.)

A **spherical triangle** is made up of three arcs
of great circles, all less than 180°.

The sum of the angles
is not fixed, but will always be greater than 180°.

If any
side of the triangle is exactly 90°, the triangle is called
**quadrantal**.

There are many formulae relating the sides and angles
of a spherical triangle.

In this course we use only two: the *sine
rule *and the *cosine rule*.

Consider a triangle ABC on the surface of a sphere with radius = 1.

*(See note)
*

We use the capital letters A, B, C to denote the
angles at these corners;

we use the lower-case letters a, b, c to
denote the *opposite *sides.

(Remember that, in spherical
geometry, the side of a triangle is the arc of a great circle,

so
it is also an angle.)

Turn the sphere so that A is at the "north
pole",

and let arc AB define the "prime meridian".

Set up a system of rectangular axes OXYZ:

O is at
the centre of the sphere;

OZ passes through A;

OX passes
through arc AB (or the extension of it);

OY is perpendicular to
both.

Find the coordinates of C in this system:

x
= sin(b) cos(A)

y = sin(b) sin(A)

z = cos(b)

Now
create a new set of axes,

keeping the y-axis fixed and moving the
"pole" from A to B

(i.e. rotating the x,y-plane through
angle c).

The new coordinates of C are

x'
= sin(a) cos(180-B) = - sin(a) cos(B)

y' = sin(a) sin(180-B) =
sin(a) sin(B)

z' = cos(a)

The relation
between the old and new systems

is simply a rotation of the
x,z-axes through angle c:

x' = x cos(c) - z
sin(c)

y' = y

z' = x sin(c) + z cos(c)

That is:

- sin(a) cos(B) =
sin(b) cos(A) cos(c) - cos(b) sin(c)

sin(a) sin(B) =
sin(b) sin(A)

cos(a)
= sin(b) cos(A) sin(c) + cos(b) cos(c)

These three equations give us the formulae for solving spherical triangles.

The first equation is the **transposed cosine rule**,

which is sometimes useful but need not be memorised.

The second equation gives the **sine rule**.
Rearrange as:

sin(a)/sin(A)
= sin(b)/sin(B)

Similarly,

sin(b)/sin(B)
= sin(c)/sin(C), etc.

So the sine rule is usually expressed
as: ** sin(a)/sin(A) = sin(b)/sin(B) =
sin(c)/sin(C) **

The third equation gives the **cosine rule**:

**cos(a) = cos(b) cos(c) + sin(b) sin(c)
cos(A) **and similarly:

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C)

Here they are together:

**sine
rule:**** sin(a)/sin(A) = sin(b)/sin(B) =
sin(c)/sin(C) cosine rule: cos(a)
= cos(b) cos(c) + sin(b) sin(c) cos(A) cos(b)
= cos(c) cos(a) + sin(c) sin(a) cos(B) cos(c)
= cos(a) cos(b) + sin(a) sin(b) cos(C)**

The cosine rule will solve almost any triangle if it
is applied often enough.

The sine rule is simpler to remember but
not always applicable.

**Note **that both formulae can suffer from
ambiguity: *E.g.* if the sine rule yields

sin(x)
= 0.5,

then x may be 30° *or* 150°.

Or, if the
cosine rule yields

cos(x) = 0.5,

then x may
be 60° *or *300° (-60°).

In this case, there is
no ambiguity if x is a *side *of the triangle, as it must be
less than 180°,

but there could still be uncertainty if an
*angle *of the triangle was positive or negative.

So, when applying either formula, check to see if the
answer is sensible.

If in doubt, recalculate using the other
formula, as a check.

**Exercise:**Alderney,
in the Channel Islands, has longitude 2°W, latitude 50°N.

Winnipeg, in Canada, has longitude 97°W, latitude 50°N.

How far apart are they, in nautical miles, along a great-circle arc?

If you set off from Alderney on a great-circle route to Winnipeg,

in what direction (towards what azimuth) would you head?

Click here for the answer.

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terrestrial sphere

Next section: Coordinate
systems: the horizontal or "alt-az" system

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