Mistake with A^2 - B^2

In algebra, a very common mistake you can see learner making is the below:

5² - 2² = (5 - 2)² = 3² = 9

This is very interesting.

It seems to be correct. That is the problem with this form of mathematical operation.

If you are aware that A² - B² = (A + B)(A - B), then this mistake will not occur.

It is this slip-of-the-mind type of human error.
It occurs when you are not alert or too tired after too many assignment quesions.

The correct answer is 5² - 2² = 25 - 4 = 21.
Or (5 + 2)(5 - 2) = 7 x 3 = 21.
Simple?

This is why maths is interesting. It catches you when you are not alert!

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Key Differences Between Cos and Log Operations

It has been mentioned in previous posts that "cos" and "log" are actually mathematical operators.

They cannot exist alone. They need partners.

For "cos", it needs angle.
For "log", it requires a number.

They operate on something.

So what is their difference since they are both operators?

"cos" changes angle information to number (ratio).

"log" operates on number to get another number.

Their applications, thus, differ in the above matter.

NOTE: "cos" here refers to any trigonometrical functions.

"cos" is useful in getting length or angle information.

"log" can be used to find the value of the power needed.
Example: To find 2^x = 5

Knowing the various mathematical tools, we can apply the correct ones to solve specific problems.

Without understanding the underlying usage and concepts, we will be at a lost to which tools to use.

Math is, therefore, wider in nature than just calculation.
Keep learning....

:-)

Math Challenge 12

Although math may seem difficult at times, proper use of its principles will render it simple to handle and use.

Here, a bit of algebraic juice can help in solving the below math challenge.

Following the pattern, you will observe a "trick" that you can make use to solve the challenge later.

2² - 1² = 3

3² - 2² = 5

6² - 5² = 11

11² - 9² = 40

The above equations are done without detailed written working, just simple mental calculation.

Challenge:
16² - 14² = ??

Do it without calculator.

What is the answer to the above? At the comment session, please.


Hmmm........ tick tock tick tock

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Understanding Math Principles Helps

There are many formulae that a student has to know and sometimes remember for his studies, and applications.

How then can he capture all these necessary formulae for usage?

One way is to understand the principles and derive those required later on for usage.

Example:

In trigonometry, you will come across sine of two angles, sin (A + B).
You have memorised sin (A + B) as "sinA cosB + cosA sinB".

However, when you further need to know sin 2A, what then?
If you know and understand the principles of the sin(A + B), you can easily move on to derive the sin2A.

How?

Since you know sin (A + B), you can equate A = B, to allow you to get the sin(A + A).
As sin (A + A) = sin2A, you will then have no problem achieving
sin(A +A) = sinA cosA + cosA sinA ==> sin 2A = 2sinAcosA

There, you have obtained another formula without the need to memorise it.

Thus mastering the sin (A + B) principles or its equivalent, can allow you to expand your knowledge further.

You see the benefits now?

I have shown only the trigonometry part in math learning, but, you can appreciate that it applies to any other topics as well.

Seek, therefore to handle principles of math well as it will serve you good in the long run.

:-) ..... :-) Forever liking mathematics. Maths is interesting!

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Math Challenge 11

What is the simplest way to solve the below equation?

cos X = log X.

You may wish to give your suggestions in the comment space.

Hints:
1) There are 3 answers to the above relation.
2) Sometimes technique not related to logarithm or trigonometry can be useful.

Happy thinking ....... (:-)

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Mistake in Cos(A + B)

We do encounter question like,

"Find the angle of A in cos (A + 45) = 0.42 ".

What do you do?

Two solutions are presented as below:

Solution 1:
cos (A + 45) = 0.42
==> A + 45 = cos^-1 0.42
==> A = 65.17 - 45 = 20.17 (Answer)

Solution 2:
cosA + cos45 = 0.42
==> cosA = 0.42 - 0.707 = - 0.287
==> A = cos^-1(-0.287)
==> A = 106.69 (Answer)

You can see that the 2 answers are different.
Why? Or is there 2 valid answers?

Looking carefully at the solutions above, you will see two concepts in approaching the solving.

The first working went through the conventional inverse cosine operation using the summed up angle (A + 45) as a piece.

The second solution used the concept of algebraic factorising to split the angles A and 45 before processing them separately.
What is wrong here?

To reveal the answer in advance, the first solution is correct while the second has a common mathematical fault.

cos (A + 45) means an operation of cosine onto the angles (A + 45) as a whole.

"cos" is not a variable to be operated upon.
Therefore, "cos" cannot be factorised!

The step, cos (A + 45), cannot be equal to cosA + cos45.
This is a common mistake that need to be removed from the brain.
Press the "Delete" button.

With this post, your trigonometry is getting better right?
Cheers!

You may visit this post for more mistakes to be avoided.

;)