Unizor - Trigonometry - Right Triangles
Channel: Zor Shekhtman on UNIZOR Education
Date: 20/06/2013

There are a few important facts about right triangles that we would like to present here.
Consider a right triangle ΔABC with the right angle at vertex C.
Let's denote sides and angles as follows.
∠BAC = α (acute angle)
∠ABC = β (acute angle)
∠ACB = γ = 90° = π/2
Cathetus BC = a
Cathetus AC = b
Hypotenuse AB = c

We will try to maintain a standard of denoting vertices with capital Latin letters, sides - with lower case Latin letters using the same letter for a vertex and an opposite side and angles - with lower case Greek letters corresponding to Latin letters for vertices of these angles. It's always a good practice to use some standard notation.

In the introductory lecture about trigonometry we introduced trigonometric functions of angles in the right triangle as ratios between its sides. The next lecture defined trigonometric functions of any angles, not necessarily acute ones, as in a right triangles, and we showed the equivalence between introductory concepts and more universal definitions. So, in this lecture, since we are discussing only right triangles, we will use the definitions of trigonometric functions as introduced via ratios of the sides of right triangles.

Using notations above, here is a list of definitions of trigonometric functions of angles in the right triangle.

sin α = a/c
cos α = b/c
tan α = a/b
cot α = b/a
sec α = c/b
csc α = c/a

Analogously,
sin β = b/c
cos β = a/c
tan β = b/a
cot β = a/b
sec β = c/a
csc β = c/b

Notice that α+β=90°=π/2 and sin α = cos β as well as cos α = sin β, which follows directly from definition.
Also worth mentioning, function tan is sometimes abbreviated as tg and function csc is sometimes abbreviated as cosec.

Out of these six functions the first three (sin, cos and tan) are used more than others with sin and cos being the most frequently used ones. The primary reason is that using just these two, sin and cos, we can express all others using obvious equalities:
tan α = sin α / cos α
cot α = cos α / sin α
sec α = 1 / cos α
csc α = 1 / sin α

Directly from definitions of these trigonometric functions follow the following equalities helpful in determining one side of a right triangle if another side and an acute angle are known.
Equalities for a cathetus:
a = c·sin α
a = c·cos β
a = b·tan α
a = b·cot β
(with similar equalities for another cathetus)
Equalities for a hypotenuse:
c = a / sin α
c = a / cos β
(with similar equalities for another cathetus)

Let's calculate the values of basic trigonometric functions (sin, cos and tan) for a few special cases of right triangles.

1. Right triangle with 45°=π/4 acute angle (another acute angle is, therefore, also 45°)
Let one cathetus be a. Then another cathetus, as can be easily shown (and proven in the Geometry part of this course), is also a and a hypotenuse, according to Pythagorean theorem, is a·√2.
Therefore,
sin 45°=a/(a·√2)=1/√2 = √2/2.
Same for cos since catheti are congruent.
tan 45° = 1 for the same reason.
2. Right triangle with 30°=π/6 acute angle (another acute angle is, therefore, 60°)
Let a smaller cathetus opposite to an angle of 30° be a. Then the hypotenuse, as can be easily shown (and proven in the Geometry part of this course), is 2·a and another cathetus, according to Pythagorean theorem, is a·√3.
Therefore,
sin 30°=a/(2·a)=1/2
cos 30°=(a·√3)/(2·a)=√3/2
tan 30°=a/(a·√3)=√3/3
Values of trigonometric functions for an angle of 60° can be obtained from corresponding values for a 30° angle:
sin 60° = cos 30° = √3/2
cos 60° = sin 30° = 1/2
tan 60° = sin 60° / cos 60° = √3