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Glossary

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Triangles and TrigonometryTrigonometry

Reveal All Steps

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 AA condition we know that they are all similarcongruent:

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:

OppositeHypotenuseAdjacentHypotenuseOppositeAdjacent

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:
    sinα=OppositeHypotenuse
  • Cosine:
    cosα=AdjacentHypotenuse
  • Tangent:
    tanα=OppositeAdjacent

COMING SOON – More on Trigonometry

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,

sinaa=sinbb=sincc

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

c2=a2+b22abcosC
b2=c2+a22cacosB
a2=b2+c22bccosA

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 supplementary angle, we know that it must be °. Now we can use the sum of the internal angles of a triangle to work out that angle β is °.

Now we know all three angles of the triangle, as well as one of the sides. This is enough to use the sine rulecosine rule to find the distance d:

sin151°d5=sin5d
d=sin151°×5sin
=23.2 km

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 oppositeadjacent side. We can find it using the definition of sin:

sin23°=height2323height
height=sin23°×23
=8.987 km

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.