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Triangles and TrigonometrySine and Cosine Rules

Reading time: ~10 min
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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.