Table Of Contents
The Simple Trigonometric Functions
Useful Trigonometric Identities
Trigonometric Functions Of
Common Angles
The mathematics used in this article is that which is taught in algebra and trigonometry classes in secondary or high schools in the USA. But realizing that the reader will most likely not have used math in a good while, I have summarized in this section the rules and laws that are used throughout the article.
Refer to Figure 2.1. In this figure is shown a right triangle. That is, the angle at B is 90 degrees. The Greek mathematician, Pythagoras, determined that in such a triangle that the square of the hypotenuse, R, is equal to the sum of the squares of the other two sides, x and y. This law can be written as:
2.1) _{}
or equivalently
2.2) _{}
A quadratic equation is one involving the square of a quantity and is said to be of degree 2. It has the following form:
_{}
The solution to this equation is given by the quadratic formula which is
2.3) _{}
Note that the equation has two solutions as shown by the presence of the ‘plus and minus sign’, _{}. Every quadratic equation has two solutions.
Again refer to Figure 2.1. The angle at the vertex C is labeled q. In a right triangle the fundamental trigonometric functions are defined as follows:
2.4) _{}
The following relationships hold in a right triangle and are generally useful:
2.5) _{}
2.6) _{}
2.7) _{}
2.8) _{}
2.9) _{}
2.10) _{}
2.11) _{}
Consider Figure 2.2. In this figure is shown an oblique triangle that is not a right triangle. The vertices are labeled A, B and C. The sides are labeled a, b and c. The angles are labeled a, b and g (the Greek letters alpha, beta and gamma). In such a triangle, the Law of Sines is stated as follows:
In any triangle, any two sides are proportional to the sines of the opposite angles.
Referring to Figure 2.2, this can be written as follows:
2.12) _{}
Again refer to Figure 2.2. The Law of Cosines can be stated as:
In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice their product times the cosine of their included angle.
In Figure 2.2, this law can be stated like this:
2.13) _{}
The following table presents a summary of the mathematical notation used in the paragraphs which follow.
Item |
Symbol |
Length of an edge |
a |
Radius of circumscribed sphere |
R |
Radius of inscribed sphere |
r |
Height of a figure |
h |
The Golden Ratio |
F |
Angles |
a, b,
g, q |
Angle q |
sin(q) |
cos(q) |
tan(q) |
0 |
0 |
1 |
0 |
30 |
_{} |
_{} |
_{} |
45 |
_{} |
_{} |
1 |
60 |
_{} |
_{} |
_{} |
90 |
1 |
0 |
undefined |
The following identity is very useful in reducing equations involving square roots to common terms:
2.14) _{} and _{} for any positive number n
Examples:
_{} and _{}
_{}
A right triangle with the sides and vertices labeled.
An oblique triangle. i.e. a triangle that is not a right triangle.
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