# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS

## CHAPTER 2, SECTION D: WORKING OUT EXACT VALUES OF THE TRIG FUNCTIONS OF THE SPECIAL FIRST QUADRANT ANGLES π/6, π/4 AND π/3 USING ELEMENTARY GEOMETRY

OVERVIEW:

This section of Chapter 2 shows how, using very elementary geometrical arguments, it is possible to construct examples of right-angle triangles with known side lengths having one of the angles π/6, π/4 or π/3 as an interior non-right angle. Having constructed these triangles, the basic geometric definitions of the six trig functions sine, cosine, secant, cosecant, tangent and cotangent as the ratios of the sides of right-angle triangles allow the exact values of the trig functions for all three of these angles to be evaluated. These values are easily reconstructed from the underlying geometry, and do not have to be memorized. The topics to be covered in this section are:

• the geometrical constructions of the sample triangles having interior angle π/6, π/4 or π/3, including how one fixes the lengths of all the sides;

• using these triangles and the ratios-of-sides-of-right-angle-triangles definitions of the six trig functions to work out the exact trig function values; and

• some helpful tips on how to simplify remembering the sample triangle with interior non-right angles π/6 and π/3.

A list of the background material needed to follow the contents of this section is given below. If you are unfamiliar with one of the background topics, or feel you would like to review it, click on the relevant link.

If you are already familar with this background information, you can proceed directly to the contents of the current section of the course by clicking HERE.

BACKGROUND MATERIAL (AND LINKS):

To follow the material in this section of the course you should be familiar with

• the geometrical meaning of angles and angular measure, as well as the basic conventions used in assigning an angle to a given direction in the plane (reviewable by reading (or re-reading) Section (b) of Chapter 1 of the course);

• the basic fact that the sum of the three interior angles of any triangle is π radians (180 degrees) (reviewable by reading (or re-reading) Section (a), of Chapter 2 of the course);

• Pythagoras' Theorem (reviewable by reading (or re-reading) Section (b), of Chapter 2 of the course); and

• the basic geometrical definitions of the six trig functions of a first quadrant angle θ as ratios of the sides of a right-angle triangle having θ as one of its non-right angles, phrased in terms of the concepts "hypoteneuse", and the sides of the triangle "opposite" and "adjacent" to the angle &theta (reviewable by reading (or re-reading) Section (c) of this chapter of the course).

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