Introducing Trigonometry

You are introduced to trigonometry in grade 10 for the first time. If I ask you to tell me your height , you will answer me by measuring it using measuring tape. But if I ask you to find the the height of a pillar , a pole or a building , all taller objects than you, so probably you would make a guess.

It is possible to measure long distances and heights using a branch of mathematics , called trigonometry.

Word trigonometry , is derived from three Greek words tri( means three) , gon (means sides) and metron (means measure). It is the study of measuring sides and angles of a triangle.

In day to day life, it is widely applied in many areas...

• all builders, architects plan the positioning of objects using trigonometry
  
• engineers and scientists use graphs of trigonometric functions in astronomical analysis and positioning of planets and stars.
  
• It is widely applied in navigation and surveying.

Using trigonometry,we can calculate the lengths of the sides and angles of a right-angled triangle, provided ,one angle and the length of one side are given, or the lengths of two sides are known to us.

In a right triangle, the longest side, which is opposite to the right angle, is called the hypotenuse. The other sides are named depending on their position relating to the angle that is to be found.

If we name an angle in a right triangle as theta, which is other than the right angle, then the remaining two sides are called side opposite of theta and side adjacent to theta. In the following figures sides are named according to the angles of the triangle.

There are 6 T-Ratios (trigonometric ratios) (For figure 1)

Sin A = BC/AC
Cos A = BA/AC
Tan A = BC/BA
Sec A = AC/BA
CosecA = AC/BC
Cot A = BA/BC

Soma Cube Puzzle

Unit cubes fun is continued...

To develop skills in visualizing spatial relationships, we in our mathematics laboratory are playing a lot with 7 pieces of soma cube (a 3 dimensional puzzle invented by Piet Hein).

The 7 pieces invented by him are shown below.

Activities :

• To make 7 pieces of soma cube puzzle using unit cubes. For doing this we use plastic cubes with grooves,which can be easily joined to each other .You may use wooden unit cubes and glue them from their faces to get the desired 7 pieces.
• To make a 3x3x3 cube using all 7 pieces .
• To make various 3 dimensional shapes using 7 pieces like a bed, a pyramid etc.

Have fun!

Interesting cube colouring problem

Do the following activity using unit cubes.

• A large cube is made from unit cubes. For example 2x2x2 , 3x3x3 etc.



• Then the outside of the big cube is painted on all six faces. You may use stickers and paste them on each face. (It would be easier for you)


• Now observe the following and note down the readings...
• How many unit cubes are painted on 0, 1, 2, or 3 faces?
  Dimension of cube ----2x2x2
  
• 0 faces painted-------- 0
• 1 face painted--------- 0
• 2 faces painted-------- 0
• 3 faces painted ------- 8
• Total unit cubes-------8 . Try making cubes of dimension 3x3x3 ,4x4x4 and 5x5x5 and do cube colouring problem using stickers.
  Observe a pattern from your readings.
• What if the base of the cube formed is not painted?

Missing unit cubes problems

Unit cubes are amazing !

In our mathematics laboratory we are using them for learning mathematics based on cubes and cuboids. Using unit cubes has not only helped in creating imaginative power in students but also developing a skill of doing mathematics by experiments and verification through hands on. This strategy has helped students in enhancing their power of visualising 3 dimensional objects.

Following are some activities which may be conducted in a mathematics laboratory using unit cubes.

Activity- Finding missing Unit Cubes

Activity Aim : To find the number of missing unit cubes to complete a cube of given dimension. Material Required : Unit cubes Procedure: Consider the given shape , as shown below

How many unit cubes are needed to complete a cube having dimension 3x3x3?

Add unit cubes and make a cube of dimension 3x3x3.

You will get a shape shown below

In this case, 3 unit cubes (1 yellow, 1 pink and 1 blue )are added to complete the cube.

Now, do this activity for the following figure to get a cube of dimension 4x4x4.

Hint:

Statistical Analysis project

Frequency of letters / words in a language

ObjectiveAnalysis of a language text, using graphical and pie chart techniques.

Description
1.Select any paragraph containing approximately 250 words fromany source. e.g. newspaper, magazine, textbook, etc.
2. Read every word and obtain a frequency table for each letter of thealphabet as follows
Letter Frequency
A
B
C
D
...
X
Y
Z
3. Note down the number of two-letter words, three-letter words, …. so onand obtain a frequency table as follows
Number of words Frequency
2 letters
3letters
4 letters
....
4. Select 10 different words from the text which have frequency greater than 1. Giveranks 1, 2, 3, …., 10 in decreasing order of their frequency. Obtain a table as follows
Word Frequency
on
it
......
5. Investigate the following
From table 1
a) What is the most frequently occurring letter?
b) What is the least frequently occurring letter?
c) Compare the frequency of vowels
d) Which vowel is most commonly used?
e) Which vowel has the least frequency?
f) Make a pie chart of the vowels a, e, i, o, u, and remaining letters. (The piechart will thus have 6 sectors.)
g) Compare the percentage of vowels with that of consonants in the given text.
From table 2
a) Compare the frequency of two letter words, three letter words, ….. and so on.
b) Make a pie chart. Note any interesting patterns.
From table 3
a) The relation between the frequency of a word to its rank.
b) Plot a graph between the frequency and reciprocal of word rank. What do youobserve? Do you see any interesting pattern?
c) Repeat the experiment by choosing text from any other language that you knowand see if any common pattern emerges.