# Triangles and TrigonometryTrigonometry

So far we have seen relationships between the **angles** of triangles (e.g. they always sum up to 180°) and relationships between the **sides** of triangles (e.g. Pythagoras). But there is nothing that *connects* the size of angles and sides.

For example, if I know the three sides of a triangle, how do I find the size of its angles – without drawing the triangle and measuring them using a protractor? This is where **Trigonometry** comes in!

Imagine we have a right-angled triangle, and we also know one of the two other angles, **α**. We already know that the longest side is called the **hypotenuse**. The other two are usually called the **adjacent** (which is next to angle **α**) and the **opposite** (which is opposite angle **α**).

There are many different triangles that have angles **α** and 90°, but from the

Since all of these triangles are similar, we know that their sides are proportional. In particular, the following ratios are the same for all of these triangles:

Let’s try to summarise this: we picked a certain value for **α**, and got lots of similar, right-angled triangles. All of these triangles have the same ratios of sides. Since **α** was our only variable, there must be some relationship between **α** and those ratios.

These relationships are the **Trigonometric functions** – and there are three of them:

The three Trigonometric functions are relationships between the angles and the ratios of sides in a right-angles triangle. They each have a name, as well as a 3-letter abbreviation:

**Sine:**sin α = Opposite Hypotenuse **Cosine:**cos α = Adjacent Hypotenuse **Tangent:**tan α = Opposite Adjacent

COMING SOON – Abbreviations: sin x, cos y

COMING SOON – Using calculators

COMING SOON – Examples

COMING SOON – Radians vs degrees, swapping the angles

COMING SOON – Rationalize the denominator.

## Inverse Trignometric Functions

COMING SOON – Inverse functions

## Sine 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*,

COMING SOON – Explanation

**Cosine Rule**

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

COMING SOON – Explanation

COMING SOON – 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.