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Question

One of the largest issues in ancient mathematics was accuracy—nobody had calculators that went out ten decimal places, and accuracy generally got worse as the numbers got larger. The famous Eratosthenes experiment, that can be found at famousscientists website, relied on the fact known to Thales and others that a beam of parallels cut by a transverse straight line determines equal measure for the corresponding angles. 

-Given two similar triangles, one with small measurements that can be accurately determined, and the other with large measurements, but at least one is known with accuracy, can the other two measurements be deduced? Explain and give an example.

The similarity of triangles gives rise to trigonometry. 

-How could we understand that the right triangles of trigonometry with a hypotenuse of measure 1 represent all possible right triangles?

Ultimately, the similarity of triangles is the basis for proportions between sides of two triangles, and these proportions allow for the calculations of which we are speaking here. The similarity of triangles is the foundation of trigonometry.

Answer 1

Small Sized Triangle

Let us consider a triangle ABC with three given sides, as shown in the attached image.

AB = 15 cm, BC = 12 am and AC = 8 cm

We can measure the three angles with the help of protector and will get the values as shown.

This triangle is small and can easily be drawn and we can measure the three angle A, B and C with the help of protector.

Note: There are three sides given so drawing the triangle is no issue and even we can measure the three angles.

A Large Size Triangle

Now when we have a very big triangle, for which it is not possible to draw on a piece of paper.

Now there are few things to understand,

Similar Triangles:

f the sides of the large triangle are proportional to the three sides of the small triangle, then the three angles of the large triagle will be the same as three angles of the small triangle.

For example we have a triangle PQR, with its side PQ = 1500 m, QR = 1200 m and PR = 800 m, then it is obvious we can't draw it.

What we can do, to model it on a small piece of paper and draw it according to a scale.

For example we chose the scale 100 m = 1 cm, then for sure the line PQ on paper will be 15 cm and actually it is 1500 m, similarly QR = 12 cm and PR = 8 cm.

Measuring the angles of small triangle will provide us the angles of the large triangle.

We have described all this based on the concept of similar triangles.

Similar triangles can be of different sizes, but their three angles will always be equal and their three sides will be proportional.

Using Laws of Sine and Cosine:

If between the three sides and three angles, we are given any two angles and one side between them, and or any two sides and the angle between them, we can use the laws of sines and cosines to determine the other three.

Images below show the two triangles and the Laws of Sine and Cosine.

Cosine law is applied, when we know any two sides and the angle between them,

Where as Sine Law can be used if we know two angles and one side between them or conversely.

Right Triangles

The third image shows a right triangle.

In this case we use Pythagorean theorem.

That is square of the hypotenuse is always equal to the sum of the squares of the other two sides.

That is a² + b² = c²

In this case knowing any of the angle other than the right angle, will help us to find the third one using the concept of sum of the three angles is 180⁰.

If the hypotenuse is 1 ( it can be 1 mm, 1 cm, 1 m, 1 km etc), then the possible measurements are as

Side a = 0.707107 m,

Side b = 0.707107 m

Then using the Pythagorean theorem we have

Hypotenuse ² = ( 0.707107)² + ( 0.707107)²

Hypotenuse ² = 0.5 + 0.5

= 1

Hence Hypotenuse = 1

A variety of triangles can be drawn with sides

a = 0.83666 cm

b = 0.547723 cm

Answer 2

I'm not sure what your question is.

You seem to have answered it yourself.

If 2 triangles are similar, the sides are in proportion.

In your example, in order to get the accurate measure of the lengths of the larger triangle, at least one of the lengths of the larger triangle must be know accurately.

Your observation is correct, that plane trigonometry depends on understanding of similar triangles.

The proof of the addition formulae can be proved geometrically (see Ptolemy's Theorem).

The laws of sines and cosines can also be derived geometrically.

Calculation of the trigonometric functions depends on the ability to calculate them from specific angles and

the use of multiple angle formulae.

I hope this answer helps your understanding..