Fundamental Concepts of Trigonometry and Euclidean Geometry

Measuring Angles

An angle can be measured in three ways:

1. In degrees: The circumference is divided into 360 equal parts. Each part represents an angle, with the apex at the center of the circle (sexagesimal measure), and is indicated by placing 1°.
2. In grads: The circumference is divided into 400 equal parts. Each part represents an angle, with the apex at the center of the circle, measuring a grad, and is indicated by placing 1^g.
3. In radians: An angle of one radian is an angle whose arc on the circumference has the same length as the radius of the circle, and is indicated by placing 1 rad.

Definition of Sine, Cosine, and Tangent

For an acute angle β in a right triangle:

• Sine (sin β): Defined as the ratio of the opposite leg to the hypotenuse.
• Cosine (cos β): Defined as the ratio of the adjacent leg to the hypotenuse.
• Tangent (tan β): Defined as the ratio of the opposite leg to the adjacent leg.

(Referring to an accompanying figure, where 'a' is the hypotenuse, and 'b' and 'c' are the legs opposite and adjacent to angle β, respectively):

sin β = b / a, cos β = c / a, tan β = b / c

The Leg Theorem (First Theorem of Euclid)

In a right triangle, the square of the length of one leg is equal to the product of the length of the hypotenuse and the length of the projection of that leg onto the hypotenuse.

Demonstration

Consider a right triangle ABC. Let $a$ be the hypotenuse, $c$ and $b$ be the legs, and $c'$ and $b'$ be their respective projections onto $a$. M is the point where the altitude from the right angle meets the hypotenuse.

In triangle ABC, cos(∠CBA) = c / a.

In triangle AMB, cos(∠MBA) = c' / c.

Since the angles are equal, we have c / a = c' / c.

Therefore, c² = a · c'.

Similarly, it is shown that b² = a · b'. (Q.E.D.)

Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

Demonstration (Proof by Area Dissection)

Take a large square with side length (b + c). This square is divided into a smaller inner square (with side $a$, the hypotenuse) and four congruent right triangles (with legs $b$ and $c$).

The area of the outer square is (b + c)².

This area is also equal to the sum of the area of the inner square (a²) and the area of the four triangles (4 · (bc/2) = 2bc).

Therefore:

(b + c)² = a² + 2bc

b² + 2bc + c² = a² + 2bc

Subtracting 2bc from both sides yields:

b² + c² = a²

As we wanted to prove. (Q.E.D.)

The Altitude Theorem (Second Theorem of Euclid)

In a right triangle, the square of the length of the altitude drawn to the hypotenuse is equal to the product of the lengths of the two segments (projections) into which the altitude divides the hypotenuse.

Demonstration

Let $h$ be the altitude to the hypotenuse, and $b'$ and $c'$ be the projections of the legs onto the hypotenuse.

By similarity, the angle ∠MBA equals the angle ∠MAC.

In the triangle corresponding to ∠MBA:

tan(∠MBA) = h / c'

In the triangle corresponding to ∠MAC:

tan(∠MAC) = b' / h

Since the angles are equal, we set the tangents equal:

h / c' = b' / h

Cross-multiplying yields:

h² = b' · c'

That is what we wanted to prove. (Q.E.D.)