06Atrigdefns

toc = Basic Trigonometry Definitions =

{“tri” =3, “gon” = angle, “metry” = measurement, so “trigonometry” = measurement of triangles}


 * Trigonometry ** is a fundamental part of many branches of science, engineering and technology.

History

 * Ancient Greek mathematicians such as Euclid(Alexandria 300 BC) and Archimedes(Syracuse 250BC) studied the geometric properties of right-angled triangles and proved theorems equivalent to modern trigonometry.
 * The modern sine function was first defined in the Indian book on Astronomy: Surya Siddhanta (age disputed) and this work was expanded on by Aryabhata (India, 500AD).
 * Tenth century Islamic mathematicianswere using essentially modern trigonometry to solve a variety of problems.
 * At about the same time, Chinese mathematicians developed trigonometry independently.

Abbreviations
Sine, Cosine and Tangent are functions (like f(x)) that take a value from a domain and return a value in their range.

Sine of q is written as **sin( q )** : domain q Î R

Cosine of q is written as **cos( q )** : domain q Î R

math \text{Tangent of } \theta \text{ is written as } \tan(\theta) \; : \; \text{domain } \theta \in r \backslash \left\{ \dfrac{\pi}{2} + n\pi \,, \; n \in Z \right\} math

Recall that **Z** is the set of all integers. Some texts use **J** for the set of integers. Both **J** and **Z** are commonly accepted.

Definitions of Sine and Cosine
Make a unit circle (radius = 1 unit) with its centre at the origin.

Draw a line segment (OP = radius) from the centre to the circumference at an angle of q (turning anticlockwise from the positive x-axis).

The x-coordinate of P is defined as x = cos( q ) The y-coordinate of P is defined as y = sin( q )

We can draw the right-angled triangle, OAP, with A on the x-axis.

We can see that the other two sides will therefore have lengths:
 * OA = cos( q )
 * AP = sin( q )

From this definition, we can see that in the second and third quadrant cos( q ) will be negative. Similarly, in the third and fourth quadrant sin( q ) will be negative.

Definition of Tangent
Make a unit circle (radius = 1 unit) with its centre at the origin.

Draw a vertical line at x = 1. The line is therefore tangent to the circle at B(1, 0).

Draw a line segment (OQ) from the centre so that Q is on the vertical line. Let q be the angle BOQ.

Notice that q is defined as the angle turning anticlockwise from the positive x-axis.

The y-coordinate of Q is defined y = tan( q ).

We can draw the right-angled triangle, OBQ.

We can see that the side lengths will be
 * OB = 1
 * BQ = tan( q )

From this definition, we can see that tan( q ) will be undefined when q = 90º, because a line drawn from O at 90º will never intersect the line x = 1.

Negative Tan
If q > 90º, we extend the line backwards through the origin until it intersects with x = 1

From this definition, we can see that in the second and fourth quadrants, tan( q ) will be negative.

Negative Angles
If we measure an angle __clockwise__ from the positive x-axis, we consider it to be a negative angle. By combining a negative angle with the definitions of sin, cos and tan, we get:

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