Trigo. TOA, CAH, SOH. Easy peasy, right?
Trigonometry is one of the big topics in mathematics in secondary school and JC. But to a student encountering these foreign terms for the first time, it can take some getting used to. So, we’ve prepared some helpful tips to help you out with the fundamentals of trigonometry!
We know things can get pretty hard in trigo – especially when it comes to the more advanced problems like proving equations. But it pays to get build a strong foundation first. If you find yourself struggling with the basics, or always making the same mistakes, read on!
Draw it out
Do you have a favourite shape? When doing trigonometry questions, your favourite shape has to be the right-angled triangle! The trigonometric ratios are based on the lengths and angles in a right-angled triangle.
If a math problem is presented without any diagrams, draw your own. When you see any diagram for a trigonometry question, the first thing you should look for are right-angled triangles. Sometimes you have to add some lines to create your own right-angled triangle.
Once you have right-angled triangles, it is easier to see where you can apply the trigonometric formulas to help you solve the problem.
Sometimes, the question may refer to a real-life application of angles, like an angle of elevation or angle of depression. Even without the question providing a diagram, you should immediately be able to draw out a diagram with the right-angle triangle to illustrate the angle of elevation or depression.
Tangent, cosine, sine don’t have units
Your math teacher may or may not have explained this to you, but tangent, cosine, and sine are ratios! This means they are describing the relationship between two things in a proportion (that is, a percentage). Remember how 50% can also be written as ½ or 0.5? Note how there are no units on the ½ or 0.5.
To illustrate this, look at the example below.
And what the equation says is basically: When the angle is in a right-angled triangle, the ratio of the opposite and hypotenuse lengths are ½. The opposite is 50% the length of the hypotenuse.
Take note of significant figures
Assuming you are a student in Singapore taking math at O level or A levels, you probably know that the standard is to leave your non-exact answers in 3 significant figures (s.f.). However, there is a special case to that: angles in degrees.
Angles in degrees should be left to 1 decimal place (d.p.) unless otherwise asked for in the question. But be careful – when the angle is in radians, the 3 s.f. rule still applies!
Now comes the problem. As tutors, we commonly see students getting an answer that is very close to the correct answer – but off by a few decimal points. A very likely reason is that those students rounded off their answers too early.
Try it! If you solve a long trigonometry question and round off all your answers in your intermediate steps to 3 s.f., you won’t get the same answer as when you only round off to 3 s.f. at the very end.
Thus, our advice is to leave your values in intermediate steps to 5 s.f. and only round it off to 3 s.f. for your final answer. This way, you won’t be letting the rounding off accumulate over the multiple steps of calculations. This is one mistake that is definitely not worth making and losing your marks over!
Pythagoras theorem is your friend
When talking about right-angled triangles, we can’t leave out a mention of Mr Pythagoras.
Note that c always refers to the hypotenuse (the longest side of the triangle), while a and b are either of the two other sides. Some questions may require the use of Pythagoras theorem to find the length of one side of the triangle, and it’s interesting how many students forget that the Pythagoras theorem exists!
Many other equations in trigonometry are also based off the Pythagoras theorem, and you can usually recognise them by the ‘squares’ in the equations.
e.g.
And there you have it! These are our tips for getting started with trigonometry, and they are sure to serve you well even when you proceed to tougher questions!
But if you need more help with trigo or other topics like basic algebra, don’t hesitate to get help directly from a tutor. While online resources can be useful for the occasional queries, having a tutor ensure you can have personal feedback on your work, and real-time suggestions on how to improve.
Here at Future Academy, we have top secondary school math tutors as well as IP math tuition for those in the IP track. Plus, you get to be guided by these tutors in a small class of not more than 6 students, meaning you’ll get plenty of attention and chance to ask all the questions you have!