01.3-TrigDefns

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.

Quadrants

Remember that we divide the Cartesian Axes into 4 Quadrants numbered 1 to 4

01.2quadnums.gif

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 $\theta$ is written as $\sin(\theta)$ : domain $\theta \in R$
  • Cosine of $\theta$ is written as $\cos(\theta)$ : domain $\theta \in R$
  • Tangent of $\theta$ is written as $\tan(\theta)$ domain $\theta \in R \backslash \left\{ \dfrac{\pi}{2} + n\pi \, , \; n \in Z \right\}$

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

01.2unitcircletrig.gif

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 $\theta$ (measured anticlockwise from the positive x-axis).

The x-coordinate of P is defined as $x = \cos(\theta)$

The y-coordinate of P is defined as $y = \sin(\theta)$

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:

  • $OA = \cos(\theta)$
  • $AP = \sin(\theta)$

From this definition, we can see that in the 2nd and 3rd quadrant $\cos(\theta)$ will be negative.

Similarly, in the 3rd and 4th quadrant $\sin(\theta)$ will be negative.

Definition of Tangent

01.2unitcircletan.gif

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 $\theta$ be the angle BOQ.

Notice that $\theta$ is defined as the angle turning anticlockwise from the positive x-axis.

The y-coordinate of Q is defined $y = \tan(\theta)$.

We can draw the right-angled triangle, OBQ.

We can see that the side lengths will be

  • $OB = 1$
  • $BQ = \tan(\theta)$

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

01.2unitcircletan2.gif

Negative Tan

If $\theta > 90^\circ$, we extend the line backwards through the origin until it intersects with x = 1

From this definition, we can see that in the 2nd and 4th quadrants, $\tan(\theta)$ will be negative.

Four Quadrant Summary

We can easily remember when to use positive or negative by the following diagram. It shows when each trig function is POSITIVE.

01.2CAST.gif

Some people use the acronym CAST to remember the table. CAST could stand for "All Stations to Croydon" or a variation on the theme

Domain and Range

01.2domaintbl.gif

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:

01.2negangletbl.gif

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