Basic Mathematics Form Four Notes – Trigonometry

Trigonometry is a branch of mathematics that deals with relationship (s) between angles and

sides of triangles.

Trigonometric Ratios

The Sine, Cosine and Tangent of an Angle Measured in the Clockwise and Anticlockwise Directions

Determine the sine, cosine and tangent of an angle measured in the clockwise and anticlockwise directions

The basic three trigonometrical ratios are sine, cosine and tangent which are written in short as

Consider the following right angled triangle.

Hypotenus: Opposite

A Adjacent

Adjacent 0 site I aa nl

cues Hypotenuse Adjancent

Also we can define the above triangle ratios by using a unit Circle centered at the origin.

x

) then the trigonometrical ratios are the same as the trigonometrical ratio of 180°-6

Y

p(x, (180 – 6)

) then the trigonometrical ratios are the same as that of 0- 180° ay

P(-x-¥

(@ —180° )

If Gis a reflex angle (270°< 6< 360"), then the trigonometrical ratios are the same as that of 360°-6

P(x,-¥) (360°- 6)

x

We have seen that trigonometrical ratios are positive or negative depending on the size of the angle and the quadrant

in which it is found.

The result can be summarized by using the following diagram.

Quadrant I! Quadrant 1

Quadrant Iv Quadrant Iii

Trigonometric Ratios to Solve Problems in Daily Life

Apply trigonometric ratios to solve problems in daily life

Example 1

Relationship between Trigonometrical ratios

Consider A ABC

[Js

A c

£

The above relationship shows that the Sine of angle is equal to the cosine of its complement.

Also from the triangle ABC above

SinA _

SinA CosA

And

2 ce

SinA CosA

Example 3

If A and B are complementary angles,

Solution If A and B are complementary angle

Example 4

For practice

Sine and Cosine Functions Sines and Cosines of Angles 0 Such That -720°<"> 720°

Find sines and cosines of angles 0 such that -720°S"> 720°

Positive and Negative angles

An angle can be either positive or negative.

Definition:

Positive angle: is an angle measures in anticlockwise direction from the positive X- axis

Negative angle: is an angle measured in clockwise direction from the positive X-axis

v From the figure above, @, is positive while 6, is negative.

Facts:

Facts:

* (90°, (-270°)]

Anticlockwise

_ (— Le —*(0,360°) v

(270°,-90°)

(180°,-180°)

a. From the above figure if is a positive angle then the corresponding negative angle to is (- 360°) or (+ – 360°)

-If is a negative angle, its corresponding positive angle is (360+) Example 5

What is the positive angle corresponding to – 46°? Solution:

-.- 460° corresponds to 314° SPECIAL ANGLES

The angles included in this group are 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°

Because the angle 0°, 90°, 180°, 270°, and 360°, lie on the axes then theirtrigonometrical ratios are summarized in the following table.

ANGLE Sine Cosine Tangent

For the angles 30° and ~~ consider the following figures.

The A ABC is an equilateral triangle of side 2 units

v3

For the angle 45° consider the following triangle

Z\

B

v

The following table summarizes the Cosine, Sine, and tangent of the angle 30°, 45° and 60°

Angle

Sine

tangent

So tan60° for example is given by ——

NB: The following figure is helpful to remember the trigonometrical ratios of special angles

from 0°to 90°

0° 30° 45° 60° 90°

2 1 0

2 3 4

If we need the sines of the above given angles for examples, we only need to take the square root

of the number below the given angle and then the result is divided by 2.

Example 7

Find the sine,cosine and tangents of each of the following angles a. -135° b. 120° c. 330°

v

Example 8

Example 9

Consider below

tan60°Sin30°

Solution:

tan60°sin30° Cos45*

_ tan60*sin30° _ Cos4S°

Exercise 2

Solve the Following.

1. find the sine and cosine of (a) 250° (b)-72° (c)-157° (d) 289°

3. Without using tables evaluates the following

The Graphs of Sine and Cosine

Draw the graphs of sine and cosine

-360° | -270° | – 180° | 270° 180°

0 1 0 -1

For cosine consider the following table of values

6 -360 | -270

Cosé |1 0

Its graph ts as follows

NAS

t

180 270 360

Each of these functions is called a period function with a period 360°

1. Usingtrigonometrical graphs in the interval -360°<0<360° Find @such that

solution

Example 10

Solution

The graphs of sine and cosine functions Interpret the graphs of sine and cosine functions

Example 11

Solution

Sine and Cosine Rules The Sine and Cosine Rules

Derive the sine and cosine rules

Consider the triangle ABC drawn on a coordinate plane

B(c,0)

From the figure above the coordinates of A, B and C are (0, 0), (c, 0) and(bCos®@, bSin@) respectively.

Now by using the distance formula

Sine Rule

Consider the triangle ABC below

From the figure above,

opposite angles"

The Sine and Cosine Rules in Solving Problems on Triangles

Apply the sine and cosine rules in solving problems on triangles

Example 12

Find the unknown side and angle in a triangle ABC given that

Solution By using sine rule, SinA Singo® _ SinB

Sinso° 8.6

7.5Sing0° _

A= 59.2° But A+ B+C=180° B = 180°-A-C =180°-59.2°-80°

SinA _ Singo* SinB

Singo° Sin40.8° 8.6 b

B.6Sin4os" _ | Sin80°

Sin26° Sins6* SinC 22.2

Sin26° __ Sing6* 222~—~—é«éi

b

_ 22.2Sin86° ~ Sin26°

_ 22.2 × 0,998 0438

Sin6s* Sin26°

22.2

Example 13

Solution

By cosine rule,

16-9-4,2025 -12.3

Example 14

Find the unknown angles in the following triangle

A

B

Solution

Using Cosine rule,

<7. 2 -48° «48

Exercise 3

3. A and B are two ports on a straight Coast line such that B is 53km east of A. A ship starting from A sails 40km to a point C in a direction E65°N. Find:

The distance a of the ship from B

The distance of the ship from the coast line. 4. Find the unknown angles and sides in the following triangle.

A

24cm

5. A rhombus has sides of length 16cm and one of its diagonals is 19cm long. Find the angles of

the rhombus.

Compound Angles The Compound of Angle Formulae or Sine, Cosine and Tangent in Solving

Trigonometric Problems

Apply the compound of angle formulae or sine, cosine and tangent in solving trigonometric

Consider the following diagram:

From the figure above

From ABCD

Bd _Ec Ab Oac'

Ee

BC

But ~ Ac

anglea with positive x-axis and OQ makes angle Pwith positive x-axes.

' From the figure above the distance d is given by

In general

v2-v6

2 2

-.Cos1 (fz + v6)

1. Withoutusing tables, find:

. Express each of the following in terms of sine, cosine and tangent of acute angles. Sin107°

Cos300°

Vectors

Displacement and Positions of Vectors

The Concept of a Vector Quantity

Explain the concept of a vector quantity

A vector – is a physical quantity which has both magnitude and direction.

The Difference Between Displacement and Position Vectors

Distinguish between displacement and position vectors

If an object moves from point A to another point say B, there is a displacement

AB which is the distance from A to B

A

<5 From the above figure A B is the Vector since it has magnitude as well as direction.

momentum, electric field and magnetic field. Other physical quantities have only magnitude, these quantities are called Scalars. For example distance, speed, pressure, time and temperature

Naming of Vectors:

Normally vectors are named by either two capital letters with an arrow above e.g.

Of, AB. etc. also a single capital letter or small letter in bold print. E.g. and sometimes a single small letters with a bar below.

y

they have different directions though the same magnitude.

Equivalent Vectors:

Suppose AB and on) are two vectors such that the length (distance) from A to B is the same as that from C to D and the direction from A to B is the same as that from C to D , then we say AB and CD are equivalent vectors.

Therefore two or more vectors are said to be equivalent if and only if they have same magnitude

and direction.

From the figure above, A\ r:8 co andE EF are equivalent vectors.

Position Vectors;

In the x —plane all vectors with initial points at the origin and their end points elsewhere are

called position vectors. Position vectors are named by the coordinates of their end points.

Consider the following diagram.

Components of position vectors:

Example 1

components

Solution:

Example 2

For each of vectors a and b shown in figure below draw a pair of equivalent vectors

Ya

Solution:

The following figure shows the vectors a and b and their respective pairs of equivalent vectors

Any Vector into I and J Components

Resolving any vector into I and J components

The unit Vectors i and j.

Definition: A unit vector is a position vector of unit length in the positive direction of x axis or y

axis in the xy—plane.

The letters iand jare used to represent unit vectors in the X axis and y — axis respectively.

Consider the following sketch,

y — axis a

xi+yj

Example 3 Write the following vectors in terms of i and j vectors:

Example 4

Write the following vectors as position vectors.

Magnitude and Direction of a Vector The Magnitude and Direction of a Vector

Calculate the magnitude and direction of a vector

Magnitude (Modules) of a Vector

Definition: The magnitude / modules of a vector is the size of a vector, it is a scalar quantity that

expresses the size of a vector regardless of its direction. Finding the magnitude of given vector.

Normally the magnitude of a given vector is calculated by using the distance formula which is

based on Pythagoras theorem.

y — axis R(0,y) — 9 P(x)

y

_— m — 0(0,0) Q(x, 0) x — axis v

Example 5

Unit Vectors:

Definition: A unit of Vector is any vector whose magnitude or modulus is one Unit.

U Now if U ts any Vector, then the Unit Vector in the direction of U is given by —— and it is

' IUI denoted by U

Example 6

Then

12 5

Direction of a vector: The direction of a Vector may be given by using either bearings or direction Cosines. (a) By Bearings:

Bearings are angles from a fixed direction in order to locate the interested places on the earth's

surface.

Reading bearings: There are two method used to read bearings, in the first method all angles are measured with reference from the North direction only where by the North is taken as 000°, the east 090°, the South is 180° and West 270°

270°W<«

180°S

From the figure above, point P is located at a bearing of 050°, while Q is located at a bearing of 135°.

Commonly the bearing of point B from point A is measured from the north direction at point A to the line joining AB and that of A from B is measured from the North direction at point B to

the line joining BA.

From the figure above the bearing of B from A, is 060° while that of A from B is 240° In the

second method two directions are used as reference directions, these are North and south.

In this method the location of places is found by reading an acute angle from the north eastwards

or westwards and from the south eastwards or westwards.

BNSO°W

C S20°E

From figure above, the direction of point A from O is N 46°E , that of B is N50°W while the direction from of C is $20°E.

Example 7

Mikumi is 140km at a bearing of 070°f from Iringa. Makambako is 160km at bearing of 215° from Iringa. Sketch the position of these towns relative to each other, hence calculate the magnitude and direction of the displacement from Makambako to Mikumi.

Solution Let lcm represent 20km, and let ABI be the displacement from Makambako to Mikumi.

The following sketch describes the location of these two places relative to each other, since AB Is the displacement from Makambako to Mikumi, then the distance between A and B is the distance between the two places.

This is to say point A stands for Makambako and B for Mikumi.

Sketch

Using the cosine rule

The displacement from Makambakoto Mikumiis 12 286kmBy sine rule

(b) Direction cosines

y — axis a

R(0,y)

+ 0(0,0) x —axis

|oP| JoP|

Example 8

— axis. Solution

Let Gand B be the angles made by a with the positive x and y axes respectively.

The direction cosines of a are 3 / 5 and 4/s

To find a we consider the direction cosine of x axis and read the angle corresponding to it from

Exercise 1

1. Find the magnitude of

P

(b) The modulus of lid + il

Sum and Difference of Vectors

The Sum of Two or More Vectors

Find the sum of two or more vectors

Addition of vectors

The sum of any two or more vectors is called the resultant of the given vectors. The sum of

vectors is governed by triangle, parallelogram and polygon laws of vector addition. (1) Triangle law of vector Addition

Adding two vectors involves joining two vectors such that the initial point of the second vector is

the end point of first vector and the resultant is obtained by completing the triangle with the

vector whose initial point is the initial point of the first vector and whose end points the end point

of the second vector.

\

From the figure above a + b is the resultant of vectors a and b as shown below

(2) The parallelogram law

When two vectors have a common initial point say P, then their resultant is obtained by completing a parallelogram, where the two vectors are the sides of the diagonal through P and

with initial point at P Example 9

Find the resultant of vectorsuandvin the following figure.

Solution

To get the resultant of vectors u and v, you need to complete the parallelogram as shown in the

following figure

Polygon law of vector addition:

If you want to add more than two vectors, you join the end point to the initial point of the vectors

one after another and the resultant is the vector joining the initial point of the first vector to the

end point of the last vector

Example 10

Find the resultant of vectors a, b, c and d as shown in the figure below.

y

In the figure above P is the initial point of a, b has been joined toaat point Q and ¢ is joined

four vectors. Opposite vectors

Two vectors are said to be opposite to each other if they have the same magnitude but different

directions

From the figure above a and b have the same magnitude (3m) but opposite direction. So a and b are opposite vectors.

Opposite vectors have zero resultant that is if a and b are opposite vectors, then

Example 11

The Difference of Vectors Find the difference of vectors Normally when subtracting one vector from another the result obtained is the same as that of

addition but to the opposite of the other vector.

Therefore the different of two vector is also the resultant vector

Consider the following figure

From the figure above

Multiplication of a Vector by a Scalar A Vector by a Scalar

Multiply a vector by a scalar

If a vector U has a magnitude m units and makes an angleOwith a positive x axis, then doubling

the magnitude of U gives a vector with magnitude 2m.

Q(x2,¥2)

Then

Example 12

Application of Vectors Vectors in Solving Simple Problems on Velocities, Displacements and Forces

Apply vectors in solving simple problems on velocities, displacements and forces

Vector knowledge is applicable in solving many practical problems as in the following examples.

A student walks 40 m in the direction § 45° E from the dormitory to the parade ground and then he walks 100m due east to his classroom. Find his displacement from dormitory to the

classroom. Solution

Consider the following figure describing the displacement which joins the dormitory D. parade

ground P and Classroom C.

From the figure above the resultant is DC. By cosine rule

«The boys' displacement from the dormitory to the classroom is 336.8 m at a bearing of 5 33° E

a. Determine the magnitude and direction of their resultant.

From the figure above, the bearing of F is N 67.4°E

Therefore the resultant force is 13 N at the bearing of N 67. 4°E

So the magnitude and direction of the force opposite to the resultant force is 13N and S67.4°W respectively…

3. Two forces acting at a point O makes angles of 30°and 135° with their resultant having

magnitude 20N as shown in the diagram below.

x and y axis respectively.

READ TOPIC 7: Matrices And Transformations