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Trigonometry

 

 

 

 

 

 

 

 

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Lesson Objectives

 

By the end of the lesson, you should be able to:

apply the knowledge of trigonometry to real life situations.

Definition of trigonometry

Trigonometric ratios

Tan: The ratio of the opposite side to the adjacent sides gives the tangent of angle

Sine: The ratio of the opposite side to the hypotenuse gives the sine of an angle

 

Cosine: The ratio of the adjacent side to the hypotenuse gives the cosine of an angle

NOTE
Each of the three ratios (sine, cosines, tangents) remain constant for a given angle regardless of the change in the lengths of the sides as shown below.

Introduction

Definition of trigonometry

Trigonometry is the study of the relationship between the sides of three triangle and their angles.
It involves the use of three ratios i.e. sine, cosine and tangents
The ratios are usually defined by the use of a right triangle as shown below

                              

Opposite: is the line opposite to the angle ?
Adjacent: is the line adjacent (next to) the angle ?
Hypotenuse: is the longest side or the line opposite the right angle (902)

 

(i)Tangent(tan ?)

The ratio of the opposite side to the adjacent sides gives the tangent of angle ?
1           
(ii)Sine (sin ?)
The ratio of the opposite side to the hypotenuse gives the sine of angle ?

2

(iii) Cosine (cos ?)

The ratio of the adjacent side to the hypotenuse gives the cosine of angle ?

3

NOTE
(a)     Each of the three ratios (sine, cosines, tangents) remain constant for a given angle regardless of the change in the lengths of the sides as shown below.

b)   Reading trigonometric tables
      The sine's cosine and tangents can be read from tables of sine cosines and tangents.

Each tables consist of three parts .the first part shows the whole number part of the angle (?)
The second part is 0.00 to 0.90 shows the first decimal part of the angle.
The part gives the mean differences

Worked examples
i) Use tables of sine's to find the sin 600

  1. Identify the angle 600   in the ? column (animate an arrow moving downward in the ? column up to the reading 600 and highlighting this value)
  1. Move along the row showing 600 and read the value below 0.00   to get sin 600

(Animate an arrow moving horizontally from 600 and another one moving from 0.00 downwards to meet at 0.8660


0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9
                                     
                                     
                                     
    0.8660                                    

The sin of 600=sin 600as 0.8660

Sin 75.64
Identify the angle 750 in the ? column (animate the arrow moving downwards in ? column) to the reading 750 and highlight this value
Move along the row showing 750   and read the value below 0.6 i.e. 0.9686
(animate the arrow moving horizontally from 750 and another moving from 0.60 downwards to meet at 0.9686)


0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9
00                                    
1                                    
2                                    
75           0.9686             2          

Proceed along the row and read value in the differences column below 4 i.e. 2

(Animate the arrow proceeding from the value 0.9686 to the value 2 in the mean difference column. Animate another arrow moving from it on the mean difference columns to meet the horizontal arrow at 2.)

Add the value 2 to 0.9686 as follows:
0.9686
+       2
0.9688
The sum of 75.64 = sin 75.64 is 0.9688

(Animate an arrow moving downwards)

  1. Use table tangents to find
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9
0                                    
1                                    
                                     
    45 1.0000                                  
    1. Tan 45

Solution

  1. identify the angle 450 in the ? column.

      (Animate an arrow moving downwards in the ? column upto reading 450 and highlight this value).

  1. Move along the row showing 450 and read the value below 0.00 to get tan 450.

(Animate an arrow moving horizontally from 450 and another one moving form 0.00 downwards to meet at 1.0000)
The tangent of 450 is 1.0000.

  1.  
    1. Tan 25.725

Solution
NB:
The tales give values of numbers to 4 significant figures. Therefore, we must first approximate 25.725 to 4s.f. as 25.73.

  1. Identify the angle 250 in the ? column.

(Animate an arrow moving downwards in the ?. Column upto the reading 250 highlight)

  1. Move along the row showing 250 and read the value 0.70 to get 0.4813.

(Animate an arrow moving horizontally form 250 and another one moving form 0.70 downwards to meet at 0.4813).

  1. Proceed along the row and read the value in the mean difference column.

Animate another arrow moving form 3 in the mean difference column to meet the horizontal arrow at 6.)

  1. Add the value 6 to 0.4813 as follows

0.4813
+       6
0.4819
The tangent of 25.725 is 0.4819

  1. Use tables of cosine to find
    1. Cos 30

Solution
(Proceed as examples 1a, and 2a above but use the tables of cosine).

    1. Cos 50.45

Solution
(Proceed as example 1b above)
The cosine of 50.4 is 0.6374
Proceeding to the mean difference the value under 5 is column is 7.

Subtract the value 7 form 0.6374 as shown below.
      0.6374
      -        7
      0.6367
The cosine of 50.45 is 0.6367
NB: For the cosine ration the values in the mean difference column are subtracted.     

    1. Use of sine, cosine and tangents ratios can be used to calculate the lengths and angles of right angled triangle as illustrated in the examples given below.

Worked examples

Sine

Find the length marked x in the figure below

                             
                             
  Calculate the size of the angle marked ? in the figure below

                       Tangents

Use the tables of tables of tangents to be calculate the size of the angle marked x in the figure below.

Find the length of the side marked 'a' in the following triangle

Cosine

    1. Use tables to find the hypothenuse y in the figure below

Use tables of cosine to calculate the size of the angle marked b

                        
Application of trigonometry to real life situations
The relationship between positions of objects can be linked by distances that form triangular shapes. The calculation of distances and angles that are formed in shuch situations may require the use of trigonometry. Some of the areas in which this knowledge is applied include building construction, surveying, navigation aviation and architecture.

The following examples show how trigonometry can be applied in real life situations.

Example 1
A vehicle moves a distance of 20km form town A to town B on a bearing of 1400.  It later moved due west to town C which is south of town A. Calculate the distance between town A and town C labeled y.

Solution
(Animate a toy vehicle moving form town A to B and then form B to C.)

From the diagram angles CAB = 1800 ' 1400
                                                   = 400
Cos 400 = Y/20
20cos 400 = y   (Multiplying both sides by 20
y = 20 x 0.7660
y = 15.32
The distance form town A to town C is 15.32km

Example 2
A building is 25m tall. A man standing 80m form the foot of the building sees an object at the top of the building. Find the angle of elevation of the object form the mans eye if he is 1.5m tall.

Solution
This information can be illustrated using a diagram as shown below.

The distance form the top to the mans eye level is    

25m ' 1.5m = 23.5m
Tan ? = 23
               80
Tan ? = 0.287
? = 16.030
The angle of elevation is 16.030

Example 3
In a shooting competition an object was placed on top of a tower 30m high to be used as a target for the bullets. The angle of elevation form the competition to the object was 460. Calculate the distance (x) covered by he bullet if the target was hit.

Solution
The information illustrated by the diagram below

From the above diagram
Sin 46= 30
               x

x sin 46 = 30    (Multiplying both sides by x)
x =  30
      Sin 46         (Dividing both sides by sin 46)

x = 30
     0.719

x = 41.7m

The distance covered by the bullet = 41.7m



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