ADVANCED MATHEMATICS FORM 5 – TRIGONOMETRY

Trigonometry is the study of angle measurement and functions that depends on angle.

The fundamental trigonometric ratios are

Sine (sin)

Cosine (Cos)

Tangent (Tan)

Others are cosecant (cosec)

Secant (sec)

Cotangent (cot )

Let θ be the angle in a right angled triangle; then we say

Sin θ

COS θ

Tan θ

And  = Cosecant θ = Cosec θ

= secant = sec θ

= Cotangent θ = cot θ

Consider a right angled triangle below

Sin θ = = ………..(i)

cos θ = = ……….(ii)

tan θ= = …………(iii)

= cosec θ= = (iv)

= sec θ = = (v)

= cot θ = = (vi)

But = tanθ

SPECIAL ANGLES

These are the angles which we can find their trigonometric ratios without mathematical tables or scientific calculators.

The angles are 0⁰, 30⁰, 45⁰, 60⁰, 90⁰, 180⁰, 270⁰, 360⁰.

Finding the trigonometric ratios for special angles.

Case 1: Consider 30⁰ and 60⁰

Here use an equilateral triangle with unit sides

That is

From AMB (right angled)

Then from the fig above

Sin 30⁰ = =

 

Case 2 Consider 45⁰

Here use are square with unit sides (1 unit)

^{That is}

^{From ABC (right angled)}

= +

^{= 1² + 1² = 2}

⁼

Then sin 45⁰ = =

Cos 45⁰ = =

Tan 45⁰ = = 1

Trigonometric ratios for 0⁰, 90⁰, 180⁰ and 270⁰ and 360⁰.

Here use a unit circle ‘Discussed also in O level’

A unit circle is a circle with radius (1 unit)

Suppose p(x,y) is a point in a unit circle

Generally in a unit circle

X = cosine value of an angle

Y= sine value of an angle

= Tangent of an angle

Angle measurement can be in two ways.

Clockwise direction (-ve angles)

Anticlockwise direction (+ve angles)
From a unit circle we use

X= cosine value of an angle

Y= sine value of an angle

Hence consider angles 0⁰, 90⁰, 180⁰, 270⁰, 360⁰ and their corresponding coordinates in a unit circle.

0^{0 means}

 

                 

360°means

Summary:-

 

The concept of picture and negative angles.

But sine function and tangent function are odd  functions

Cosine function is an even function

Fig above
From Sin θ =
Sin ( -θ) = – = -sinθ

cos θ = 

cos(-θ) = Cos θ

THE IDEA OF QUADRANTS

The idea is discussed in O’Level form IV Basic Mathematics, but let us recall the idea.

1^st Quadrant Angles

The range of the angles is 0°< θ<90⁰

The all trig ratios are positive and are obtained directly from four figure (mathematical figure

2nd Quadrant angles

The range of the angles is 90⁰ < θ <180⁰

3^rd Quadrant

Ranges from 180°< θ< 270°

4^th Quadrant

Ranges from 270°<θ<360°

 

Eg: Sin 315° = -Sin (360° -315°)

=-

= -tan (360° – 330°

  = -tan 30°

=

=

= =

PYTHAGORAS THEOREM (IDENTITY)

Consider a right angled

From Pythagoras theorem

+ b² = c²

Dividing by C²

+ =

+ ( = 1————–∗

Substitute equations (i) and (ii) into (*)

Then we get

Is the Pythagoras Identity.

Dividing equation (1) by

 

 dividing equation (i) by Sin²θ