Unizor - Trigonometry - Use in Geometry
Channel: Zor Shekhtman on UNIZOR Education
Date: 2013-11-25

For the purposes of presenting different problems in this and following lectures let's denote elements of any triangle by the following letters:

A, B, C are vertices;

a, b, c are sides correspondingly opposite to these vertices;

p is half-perimeter, that is p=(a+b+c)/2;

α, β, γ are angles corresponding to vertices A, B, C;

ha, hb, hc are altitudes to corresponding sides a, b, c;

ma, mb, mc are medians to corresponding sides a, b, c;

la, lb, lc are angle bisectors of corresponding angles α, β, γ;

R is a radius of a circumscribed circle;

r is a radius of an inscribed circle;

s is an area of a triangle.

Two examples of using trigonometry in geometry we've already presented in the "Simple Trigonometric Identities" chapter of this course. They are the Law of Sines and the Law of Cosines. They are quite fundamental in many geometric problems involving triangles.
For instance, from the Law of Sines sin(α)/a=sin(β)/b=sin(γ)/c we can find two sides of a triangle knowing the third side and two angles.
From the Law of Cosines c²=a²+b²−2ab·cos(γ) we can find angles of a triangle knowing its sides. Actually, we find the cosine of an angle first and, since the function y=cos(x) is monotonic on an interval (0, π) where angles of any triangle take the values from, this uniquely identifies the angle by knowing the sides.

In this lecture and the following it lectures with problems we will present a few other cases of solving geometric problems using the trigonometry.

Problems

1. Prove that for any triangle the following equality is true.
(a−b)·(a+b)·sin(γ) = c²·sin(α−β)

2. Express a, b, c in terms of R, α, β, γ and relate it to the Law of Sines.

Answer:
a = 2R·sin(α)
b = 2R·sin(β)
c = 2R·sin(γ)

3. Let H be a point of intersection of altitudes in the triangle ΔABC.
Prove that three circles that circumscribe triangles ΔABH, ΔBCH and ΔCAH have the same radius.

Hint: Use the previous problem.

4. Express s (area of a triangle) in terms of R, α, β, γ.

Answer:
s = R²·[sin(2α)+sin(2β)+sin(2γ)]/2