Basically, trigonometry is divided into 2 different levels of study ie Basic Trigonometry for PMR and Advanced Trigonometry for SPM level. Basic level will cover certain areas such as Sine, Cosine and Tangent of which the degree for each angle shall be less than 90°. On the other hand, students will be introduced to Sec, Cosec and Cot with the total degree of more than 90° in the following level.
Now let us look at how basic trigonometry works.
A Sin X = B —-> Sin X = B/A —-> X = Inverse Sin (B/A)
Since Sin 30° = 0.5, as such 2 Sin 30° shall be calculated as 2 (0.5) = 1
If 2 Cos X = 1.7321
Cos X = 1.7321/2 = 0.8661
X = Inverse Cos 0.8661 = 30°
A Sin X + A Cos Y —-> A (Sin X + Cos Y)
4 Sin 30° – 2 Cos 60° = 4(0.5) – 2(0.5) = 2 – 1 = 1
2( 2 Sin 30° – Cos 60°) = 2( 2(0.5) – 0.5 ) = 2(1 – 0.5) = 2(0.5) = 1
Find the value of M when 6 Sin 40° + 3 Cos M = 6
6 (0.6428) + 3 Cos M = 6
3.8568 + 3 Cos M = 6
3 Cos M = 6 – 3.8568 = 2.1432
Cos M = 2.1432/3 = 0.7144
M = Inverse Cos 0.7144 = 44.41° or 315.59° *
3( 2 Sin 40° + Cos M ) = 6
2 Sin 40° + Cos M = 6/3 = 2
Cos M = 1 – 2 Sin 40° = 2 – 2(0.6428) = 2 – 1.2856 = 0.7144
M = Inverse Cos 0.7144 = 44.41° or 315.59° *
* refer to the diagram below
As we have learnt from Polygons in Form 3 Maths, an individual angle may go beyond 90°. Due to this nature, we can safely say that each angle may vary from 0° up to 360° and anything more than that shall come in a repeated sequence accordingly; for example 450° is the same with 450° – 360° = 90° and so on. Now observe the diagram below that covers the range between 0° up to 360° which is defined under 4 different segment ( 360°/90° = 4) that shall be known as quadrant.
Linear Equations is another branch of Algebra and this topic is one of the most regularly used method in Maths or Add Maths. I’ve prepared few sets of examples for you to understand better the concept of Linear Equations and its application. However, before you go further into the examples shown, the following are the few basic concepts of Linear Equations that are worth remembering.
a + b = c ; a = c – b
a – b = c ; a = c + b
a ÷ b = c ; a = c x b
a x b = c ; a = c ÷ b or c x 1/b
a/b = c/d ; a = (c x b)/d
a² = b ; a = √b
³√a = b ; a = b³
- a + b – c = – d can also be written as a – b + c = d
Factorisation is one of the most important concept in Maths. It is normally used to find the roots of the equation. However, this concept is also applicable in linear and simultaneous equations. In logarithm, this concept is widely used as it works relatively well with indices. As such, factorisation can appear anywhere within the concept of Mathematics solution.
Now let us look into few examples of factorisation that is commonly used in todays syllabus.
2x + 14 = 2(x + 7)
4x² – 9y² = 2²x² – 3²y² = (2x + 3y)(2x – 3y)
16n² + 9m² = 4²n² + 3²m² = (4n + 3m)² – 24mn
Linear Equations is actually derived from Algebra and to most, it sounds more like bad news. Funny indeed as during my time as a student, this particular chapter used to be the only avenue that could offer me a lot of free marks. Bragging? No…it’s a non-fiction tale to tell. Linear Equations is so easy that the only solution is about reversing the whole operation, nothing can be simpler than that.
A lot of students ask me on the steps that need to be taken to solve the linear equations…my reply to them; there aren’t any particular steps coz there are too many ways of solving it. Let us review on some examples that I consider worth exploring.
As we can observed, linear equations can appear an re-appear in any other chapter. Its a reminder actually…as there will be no end to linear equations!!!!
To most students, it will surely be a big relief if Maths doesn’t come with any formulae at all. Ask any student and that will be the answer…for sure!!! It is not that the student doesn’t like formulae, but formulae comes in too many forms and it seems rather impossible to memorise it all. To those who have no ‘love affair’ with Maths…formulae has always been the reason, in other words, formulae is taboo to some….
Memorising formulae will never solve your problem, beside it being too many, we also need to memorise other things (far more important, depending on how you rate it yourself)…anniversary date, girlfriend/boyfriend’s date of birth, potential girfriend/boyfriend phone no and email addressess and many more…formulae is not welcome at all in our brain, not by a single-tiny percent…
If so, then how do we solve this formulae matter? Do we really need to think of it? Maybe…YES..since SPM is drawing near and for those who just cleared the PMR, you will be meeting more and more formulae in years to come.
My only solution is…try to understand the formulae. It helps me a lot those days and I’m pretty sure it will help you as well. For example, let us start with a simple Maths formulae….say, Surface Area of a cylinder. Observe the diagram below properly and you will find that a cylinder is in fact comprises of 2 segments;
- a circle as its base and top; and
- a rectangle/square as its body
- The Area and a circumference of a circle = πr2 and 2πr respectively
- The length of the rectangle/square is in fact the same with the circumference of the circle
- The breadth/width of the rectangle/square is the same with the height of the cylinder
- The Area of the rectangle/square will be 2πr x h (length x breadth/width)
As such, the total Surface Area of the cylinder can be calculated as follows:
AREA OF THE 2 CIRCLES + AREA OF THE RECTANGLE/SQUARE, that could also be written as : πr2 + πr2 + 2πrh = 2πr (r + h)
Try on other formulae as well, perhaps it could help you on how to apply basic and simple formulae for the harder ones without having to memorise it too much…after all, we need our brain for something else…
This chapter is very much ‘alive’ for those who are seating for the coming SPM…eventhough not much focus being made on it at PMR level. For those who wish to master this chapter, please take note on the following factors :-
a. Understand the basic concept of equation
- Equation of straight line is represented by y = mx + c; where m = gradient and c = y intercept
- Equation of curve is represented by y = ax² + bx + c
- NOTE THAT y OR/AND x CAN BE REPRESENTED BY OTHER FORM SUCH AS f(x), log y etc
b. Understand the 4 basic ways of finding gradient
- m = difference of y / difference of x when minimum of 2 coordinates are known
- differentiate the equation by way of dy/dx when only 1 coordinate is given (this step is learnt in Add Maths)
- gradient for perpendicular line is calculated as -1/ gradient of ordinary straight line (m1 x m2 = -1)
- gradient of normal = gradient of perpendicular line
c. Know the shape of equations
- +ve or -ve gradient is reflected differently when plotted into a graph (y = +ve or -ve mx + c)
- +ve or -ve value of a is reflected differently when plotted into a graph (y = +ve or -ve ax² + bx + c)
Bear in mind that this chapter may also require knowledge and understanding from other chapters such as logarithm, algebra, coordinate geometry, trigonometry and many more.
Basic function of straight line is defined as y = mx + c where m is the gradient and c is the Y intercepts
Gradient is calculated when minimum of 2 coordinates are given by way of placing the difference of y’s over the difference of x’s
Example : (x₁ , y₁) (x₂ , y₂)
y₁ – y₂ / x₁ – x₂ = gradient
What is c in the straight line equation of y = mx + c?
• c is the value of y on the Y axis where the line intersect with the Y axis
• Coordinate (0, c) or Y intercepts
During my days as a student, algebra always seemed to be one of my favourite topics. I found it very straight forward, easy to understand… so long as you have a strong foundation in basic Maths. In algebra, students must be dared enough to adopt self experiment and apply bracket whenever necessary. This will help a lot…. just take this piece of advice whenever you come across problem that requires algebra as a way of solution.
Today, I’m going to share with you amongst the common problems faced by most students when come to algebra.
The most popular ones is failure to expand the equation properly. Clear example is as follows :
Another common mistake is failure to differentiate between one equation to another. For example;
x² – y² IS NOT THE SAME WITH ( x – y )²
In cases like this, students are encouraged to do their own experiment; by way of replacing x and y with simple digits. For instance, let x = 5 and y = 2
x² – y² shall now read as 5² – 2² = 25 – 4 = 21
Now compare with the other equation
( x – y )² = ( 5 – 2 )² = (3)² = 9
From the experiment above, it shows clearly that
x² – y² IS NOT THE SAME WITH ( x – y )²
To master this topic will not take too much of time, just put extra effort on exercise and if there is any particular tips that ought to be given attention…that has to be
x² – y² = ( x + y )( x – y )
or 16x² – 4y² = 4²x² – 2²y² = (4x + 2y )( 4x – 2y )
x² + y² = ( x + y )² – 2xy