Right Triangle Trigonometry

Right Triangle Trigonometry

This document is a comprehensive guide on Right Triangle Trigonometry authored by Hidegkuti. It covers fundamental concepts and definitions related to trigonometric functions, including sine, cosine, and tangent. The document is structured into several parts, starting with basic definitions and moving into exact values for famous angles such as 30°, 45°, and 60°. It also includes proofs of key identities, applications in various fields, and sample problems with solutions. The guide is intended for students and educators looking to deepen their understanding of trigonometry. The content is well-organized, featuring examples, discussions, and practice problems to reinforce learning. This educational resource is suitable for both classroom use and self-study.

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Lecture Notes Right Triangle Trigonometry page 1
Part 1 - The Definitions
We will now define some fundamental concepts of trigonometry. Let α be an acute angle. (An angle α is acute
if 0 < α < 90
◦
).
Let us draw a right triangle that also contains α as an angle. Let us locate the angle α. The longest side, the hypotenuse is
always opposite the right angle. To the other two sides we will refer to as the side opposite α and the side adjacent to α. It
is important to understand that ’opposite’ alone makes no sense in this context. It is opposite the angle. And if we change
the location of the angle, that will result in changes in what sides we call what. Suppose that the red angle is labeled α and
the blue angle is labeled by β.
The horizontal size is adjacent to α but opposite to β. The vertical side is opposite to α but adjacent to β.
Suppose we measure all three sides. The following trigonometric
values belonging to angle α are defined as shows below.
Sine of α is the ratio of the lengths of two sides: the side opposite
α, divided by the length of the hypotenuse.
sin α =
length of the side opposite α
length of hypotenuse
=
a
c
Cosine of α is the ratio of the lengths of two sides: the side adjacent to α, divided by the length of the hypotenuse.
cos α =
length of the side adjacent to α
length of hypotenuse
=
b
c
Tangent of α is the ratio of the lengths of two sides: the side opposite to α, divided by the length of the side adjacent to α.
tan α =
length of the side opposite α
length of the side adjacent to α
=
a
b
These three are the most important ones, you must memorize these definitions. There are three additional definitions, but
their significance will be more obvious when studying calculus. In the mean time, remember these in terms of sine, cosine,
and tangent. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.
© Hidegkuti, 2022 Last revised: February 2, 2023
Lecture Notes Right Triangle Trigonometry page 2
Cosecant of α is the reciprocal of sin α.
csc α =
length of hypotenuse
length of the side opposite α
=
c
a
Secant of α is the reciprocal of cos α.
sec α =
length of hypotenuse
length of the side adjacent to α
=
c
b
Cotangent of α is the reciprocal of tan α.
cot α =
length of the side adjacent to α
length of the side opposite α
=
b
a
Discussion:
1. The definitions are sort of vague. Do they uniquely determine these values? Suppose
that an acute angle α is given. Ann and Bryne both draw a right triangle with α in it,
measure the sides and compute sin α, cos α, and so on. But Ann draws a small cute
triangle while Bryne draws a huge triangle. The sides are clearly different. What if sin α
is a different number based on Ann’s triangle than sin α based on Bryne’s triangle? The
would mean that sine of an angle is not well-defined. Well?
2. Prove that for any acute angle α, both sin α and cos α are positive, a real number (i.e.
unit-less), and the values are always less than 1.
3. Prove that any positive number can be a tangent of an angle. Can you draw a right
triangle that would lead to tan α = 100?
Example 1. Consider the right triangle with sides 3, 4, and 5 units long. Find all trigonometric values of the second largest
angle in the triangle.
Solution: The second largest angle is opposite the second longest side, 4 units long.
sin α =
length of the side opposite α
length of hypotenuse
=
4
5
cos α =
length of the side adjacent to α
length of hypotenuse
=
3
5
tan α =
length of the side opposite α
length of side adjacent to α
=
4
3
csc α =
5
4
sec α =
5
3
cot α =
3
4
© Hidegkuti, 2022 Last revised: February 2, 2023
Lecture Notes Right Triangle Trigonometry page 3
.
Theorem: For any acute angle α,
(sin α)
2
+ (cos α)
2
= 1
Before we prove the statement, let us introduce some new notation. Instead of (sin α)
2
, we will write sin
2
α
Using the standard notation, sin α =
a
c
and cos α =
b
c
. Then
sin
2
α + cos
2
α =
a
c
2
+
b
c
2
=
a
2
c
2
+
b
2
c
2
=
a
2
+ b
2
c
2
=
c
2
c
2
= 1
At the core of this proof was the Pythagorean theorem. We will see that trigonometry is especially rich in identities. This
one is the first of many to follow and also, probably the most fundamental identity. It is called the Pythagorean identity.
Part 2 - Exact Values
The exact value of trigonometric functions of most angles can not be determined by elementary techniques. At this point,
we are to imagine mathematicians drawing right triangles and measuring sides to obtain approximate values. There are
a few angles, however, that are exception to this; we can compute the exact values of trigonometric functions. Certain
symmetries and the Pythagorean theorem enables us to do that, in case of 30
◦
, 45
◦
, and 60
◦
.
We start by drawing an equilateral triangle
with sides 2 units long. All three angles of
this triangle measure 60
◦
.
Let D be the midpoint of side AB. We connect points C and D.
Because the triangle is isosceles, this line is perpendicular to the
base AB and cuts the triangle into two congruent right triangles.
Consequently, the two angles created at point C both measure 30
◦
.
© Hidegkuti, 2022 Last revised: February 2, 2023
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End of Document
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