# Triangles and TrigonometrySine and Cosine Rules

So far, all you’ve learned about Trigonometry only works in right-angled triangles. But most triangles are not right-angled, and there are two important results that work for all triangles

**Sine Rule**

In a triangle with sides *a*, *b* and *c*, and angles *A*, *B* and *C*,

**Cosine Rule**

In a triangle with sides *a*, *b* and *c*, and angles *A*, *B* and *C*,

COMING SOON – Proof, examples and applications

## The Great Trigonometric Survey

Do you still remember the quest to find the highest mountain on Earth from the introduction? With Trigonometry, we finally have the tools to do it!

The surveyors in India measured the angle of the top of a mountain from two different positions, 5km apart. The results were 23° and 29°.

Because angle α is a

Now we know all three angles of the triangle, as well as one of the sides. This is enough to use the *d*:

There is one final step: let’s have a look at the big, right-angled triangle. We already know the length of the hypotenuse, but what we really need is the *sin*:

And that is very close to the actual height of Mount Everest, the highest mountain on Earth: 8,848m.

This explanation greatly simplifies the extraordinary work done by the mathematicians and geographers working on the Great Trigonometrical Survey. They started from sea level at the beach, measured thousands of kilometers of distance, built surveying towers across the entire country and even accounted for the curvature of Earth.