Math Is Fun Forum

To calculate the angles of a triangle using basic geometry and the Pythagorean theorem without using sine and cosine, we can use the Law of Cosines. The Law of Cosines allows us to find an angle of a triangle when we know the lengths of all three sides.

The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and angle C opposite side c:

cos(C) = (a^2 + b^2 - c^2) / (2 * a * b)

Using this formula, let's take a simple example:

Consider a triangle with sides of length 3 units, 4 units, and 5 units.

To find angle C (opposite the side of length 5 units), we can apply the Law of Cosines:

cos(C) = (3^2 + 4^2 - 5^2) / (2 * 3 * 4)
cos(C) = (9 + 16 - 25) / (24)
cos(C) = 0 / 24
cos(C) = 0

Now, we need to find the angle C. To do that, we need to find the inverse cosine of 0 (cos⁻¹(0)).

cos⁻¹(0) = 90 degrees

So, angle C is 90 degrees.

For the remaining angles, we can use the Law of Sines or the fact that the sum of the angles in a triangle is always 180 degrees. Since we already found angle C as 90 degrees, the sum of the other two angles is 180 - 90 = 90 degrees.

Suppose the other two angles are A and B.

A + B = 90 degrees

For a simple example, let's assume angle A is 30 degrees. Then,

A + B = 30 + B = 90 degrees
B = 90 - 30 = 60 degrees

So, in this example, the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees.

By using the Law of Cosines and basic geometric principles, ancient mathematicians could calculate angles in triangles without relying on trigonometric functions like sine and cosine.

I hope this example helps you understand how triangles were calculated in ancient times. If you have any further questions or need more examples, feel free to ask!

Regards.