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Showing posts with label Mathematics. Show all posts
Showing posts with label Mathematics. Show all posts

Mathematics | Trigonometry 3 | English Medium

8:34 AM | තියන ජාති:



Through today's lesson we are going to study a little bit more deeper about trigonometry more than the last two lessons. Lets see an easier way to remember the main nine trigonometric ratios we discussed in the last article.
All of you now know that the tan ratios of an angle is equal to the ratio between, the sin ratio of that angle and the cos ratio of it. Which means if we know the sin and the cos values of those three main angles we can derive tan value using them.



If you can remember sin30, sin45 and sin60 then inverse the values of them to get cos30, cos45 and cos60. Then you can derive tan30, tan45 and tan 60 from them.

Lets do some example sums to get a better understanding about the type of questions you are going to get for the O/L examination.

Find the values of sin30.sin60 + cos30.cos60.


sin30.sin60 + cos30.c0s60 = (1/2)(√3/2) + (√3/2)(1/2)
= √3/2


I'll leave this sums to try them yourselves.

1. sin60.sin30 + cos30.cos60
2. sin30cos60 + cos30sin60
3. (sin30)^2 + (cos30)^2
4. (cos30 – sin30)(cos30+sin30)
5. Prove that (tan30 + tan45)/(1-tan30.tan45) = 2 +√3


When I was doing my O/Ls I always had a nagging question about trigonometry, why do we learn trigonometry? To get a clear answer to that question you should learn trigonometry furthermore but for now I will explain it with a simple example.



Imagine there is a somewhat tall rock and we want to measure its height. Here is a simple way of doing it. Assume that the earth is plane considering to the height of the rock. Take a rope and measure its length. Tie a rope to the point where you want to measure the height (point A). Tie the other end to the ground and keep in mind to keep the rope straight. Then measure the angle of the rope with the ground as shown in the figure below. Now we know the AB length we know the required angle so the height of the rock = AB length X sin(measured angle) 



Written by Uncle 

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Posted by Yaz

Mathematics | Trigonometry 2 | English Medium

2:39 AM | තියන ජාති:

From the previous lesson we learnt about the three main trigonometrical ratios. Lets see how we can find the values of trigonometrical ratios of some commonly used angles.



The triangle above is an equlateral triangle which means all three sides are equal in length. So AB = AC, hence ABC angle is equal to ACB angle. AC = CB, hence CAB angle is equal to CBA angle thus all three angles are equal. Let the value of an angle be x, 

 then x + x + x = 180°

 So the ABC angle is 60 degrees whilst the AN perpendicular drawn to BC from A eually divides the BAC angle so that, BAN angle is 30 degrees.
Let AB = BC = CA = 2a then,
BN = a and
AN =  √3a , according to the pythagoras theorem (apply pythagoras theorem to the triangle ABN) .

Now from BAN triangle  we can see than sin BAN is equal to the ratio, BN/BA.

Thus sin30 °  = a/2a = 1/2
Likewise we can develop the sine, cosine and tan values of 30° and 60°.
Now lets consider a right angled isoceles triangle.

Let the two equal sides be of length a then according to the pythagoras theorem the hypotaneuse of the triangle will be square root(2)a.

So easily we can see that 
sin45°  = cos 45° = 1/√2 
and 
tan 45 °  = 1.


The table below will sum up all the trignometrical ratios we have considered above and you are expected to memorize them for the O/L exam.



From above table we can roughly deduce that from 0 - 90 °  the sine and tangent values of an angle increases with the angle while the cosine value of an angle decreases with the angle.
Moreover,

sin 0 = 0   sin 90 °  = 1 

cos 0 = 1  cos 90 ° = 0
tan 0 = 0  tan 90 ° = infinity

therefor for any given angle x < 90(degrees) ,    
0 < sinx < 1
0 < cosx < 1
tanx > 0



Written by : Uncle

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Posted by Janitha

Mathematics | Trigonometry 1 | English Medium

11:02 AM | තියන ජාති:


Trigonometry is a branch of pure mathematics which goes long way back in history and which is also utilized in complex mathematical concepts that incluede geometry.

For the O/L mathematics syllabus we only have the basics of trigonometry.

First of all lets identify the 3 main trigonometrical ratios. They are,
1.) sine
2.) cosine
3.) tangent

For the O/L exam we only have to learn about these ratios about a right angled triangle.



In the diagram show above the angle C is denoted by theta.
In the O/L syllabus the definitions of trigonometrical ratios are simplified so that the students could get a better understanding about them.
For the angle theta the trigonometrical ratios can be defined as follows,



Generally we can identify that sin ratio of an angle is the ratio between the length of the oppsite side and  the hypotneuse relative to the angle. The cosine ratio of an angle is the ratio between the length of adjacent side and the hypotaneuse relative to the angle. The tan ration of an angle is the ratio between the length of opposite side and the adjacent side relative to the angle.


We can see that, 

therefore,



Written by : Uncle


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Posted by Janitha
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