TRIGONOMETRY

Introduction

The word trigonometry is derived from two Greek words trigono-triangle and metria means “measurement of triangles” Initially, trigonometry dealt with relationships among the sides and angles of triangles, and was used in the development of astronomy, navigation and surveying.

However, it is worth noting that with the development of calculus and physical science in the 17^th century, a different perspective arose one that viewed classic trigonometric relationship as functions with the set of real numbers as their domains. As a result of this, the application of trigonometry expanded to include a vast number of physical phenomena involving rotations, or vibrations, soundwaves, light rays,planetry orbits vibrating strings, orbit of atomic particle.

To bring Trigonometry to its present modern form many great mathematician have contributed immensely. We mention Ptolemy (90-160), Copernicus (1473-1543) the Swiss mathematician Leonhard Euler (1707-1783)

 

ANGLES

An angle is the amount of turning or rotation of a line about one of its extremities in a plane from one position to another.

The starting position before the turning or rotation is the initial side of the angle, and the position after rotation is the terminal side as shown in diagram below. The extreme end or endpoint of the turning is known as vertex of the angle.

Vertex

 

 

 

 

	Fig 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig 2, shows an angle in its standard position. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation as shown in fig 3. We use Greek letters

 

 

 

MEASUREMENT OF ANGLES

When measuring angles a particular angle is fixed and is taken as a unit of measurement , so that any other angle is measured by the number of times it contains the unit.

 

SYSTEM OF MEASUREMENT

There are three major system of measurement. They are:

1.      Sexagesimal System (English system)

2.      Centesimal System (French System)

3.      Circular System

1. Sexagesimal System:- The unit of measurement in the system is degree, This is so called because each unit is divided 60 parts (Sexagesimus means sixtieth)

1 right angle = 90 degree

1 degree = 60 minutes

1 minute = 60 seconds

1 Degree, 1minute and 1second are written as

2. Centesimal System:- This is the French system of measuring angles. Each unit is divided into 100 equal parts called “grades”. Each grade is divided into 100 equal parts called minutes and each minute is divided in 100 equal parts called seconds.

1 right angle = 100 grades

1 grade =100 minutes

1 minute = 100 seconds

a grade, a minute and a second is written as

          

3.         Circular system: - In this system, the unit of measurement is Radian. One    radian is the measure of central angle  that subtends an arc s equal to the radius r of the circle.

 

 

 

 

 

 

 

 

 

 

Arc length = radius when

The circumference of a circle is given as , it follows that a central of one full revolution (counter clockwise) corresponds to an arc length of  s = . Also each radian intercept an arc of lengtb r, we thus conclude that one full revolution correspond to an angle /r =  radian.

In general, the radian measure of a central angle  is obtained by dividing the arc length s by r. that is

 

Relation between the Sexagesimal system and Circular system

Since  is the measure of an angle of one complete revolution, degrees and radians are related by the equations

Example

Convert the following radian to Degree

 

TRIGONOMETRIC FUNCTIONS AND RIGHT TRIANGLES.

Given the right-angled triangle ABC show in the figure below

 

 

 

 

 

 

 

 

The three sides of the right angles are labeled Hypotenuse AB, the opposite (AC) {the side opposite to the angle  }and the adjacent side {the side adjacent to the angle  } BC

We define the trigonometric ration as follows:

Example 1: Find the value of the six trigonometric ratio of  as shown in the figure below

 

 

 

 

 

 

 

 

Solution

By Pythagoras theorem

 

Thus, we have adj =12, opp = 5, hyp= 13, therefore the six trigonometric ratio are as follows:

 

Solution

 

 

 

 

By Pythagoras theorem, the length of the third side is BC =                 

   

Example: Given that  is an acute and that sec  =

 

 

 

 

 

 

 

 

 

By pythagora’s the length of the third side that is AC=

                       

 

EVALUATING TRIGONOMETRIC RATIO OF

The ratio of 45^o

Draw an isosceles right angle ABC,then angleA=angle C=45^o_, the size of the triangle will not affect the ratio of 45^o_, so let AB=AC=1cm and then by Pythagoras BC=

                 

 

 

 

 

 

              

From definition

The ratio of 30^o and 60^o

Construct an equilateral triangle. For ease choose each side to be x cm. Draw the perpendicular AD from A to B

 

 

 

 

 

 

 

By Pythagoras AD=

From the definition

 

 

 

 

 

 

 

TRIGONOMETRIC IDENTITIES AND EQUATION

PYTHAGOREAN IDENTITIES

 

 

 

 

 

 

 

The figure above shows a unit circle  OAB is a a right-angled triangle  with OB=1{radius} OA=x, and AB=y 

 from the definition of trigonometric

 ratio.

x=cos                       ………(i)

y=sin                         ……….(ii)

from (i) x²=  cos²        ………..(iii)

from (ii) y²=  sin^{2           ……………..(iv)}

Adding (iii) and (iv)

 

 

 

Example

Verify the following identities

 

Example

 

 

 

 

 

 

 

 

 

SUM  AND DIFFERENCE FORMULA

In this section, we shall study some important identities (or formula) and how they can be employed to simplify or change trigonometrical expression. We shall not delve into proving some of these identities. We begin with eight sum and difference formula that expresses trigonometric function of (A  B) as function of A and B alone.

 

 

 

 

 

Example