3D trigonometry: can't see which triangle to use? Three steps to extract the plane triangle
The difficulty in 3D trigonometry is usually not the formula but choosing the diagram: the question gives you a 3D figure, but the formula only applies to a plane triangle. A practical approach is first to identify every right angle in the figure and find the sides whose length or angle is known, then extract one triangle at a time, writing its three sides and included angle on paper, and only start calculating once you have confirmed there is enough known information; a single question often requires two or three triangles to be extracted and handled separately. When revising, get into the habit of drawing the side view and plan view first, and state the reason for each step.
This article was put together automatically from general study experience and is for reference only. For exam and school arrangements, check the latest information from the organisation concerned.
- The formulas of 3D trigonometry only apply to plane triangles; the difficulty is choosing the diagram, not calculating.
- First mark the right angles and known quantities, then decide which triangles to use.
- Extract only one triangle at a time and draw it separately beside the figure.
- State the reason at each step, and put intermediate results back into the 3D figure before extracting the next triangle.
- When revising, you can practise only the first step: extracting the diagram and planning the order.
Where the difficulty lies: there is no ready-made formula on a 3D figure
There are not many formulas in 3D trigonometry itself; it comes down again and again to the ratios of sine, cosine and tangent. The real difficulty is this: the formulas only hold inside a plane triangle, whereas the question gives you a three-dimensional figure, and the relationships between sides are hidden in different planes. Staring at the 3D figure, students often cannot even see clearly which side truly meets which side, so they do not know which set of data to substitute.
Another common situation: a question actually needs two triangles, the first to find an intermediate side and the second to find the answer. If you try to calculate straight away in the 'most direct' triangle, you will find there is not enough known information. This is not a calculation error but the wrong order of attack. So the first step in 3D trigonometry is not to write down a formula but to plan which triangles to use.
Step one: mark all right angles and known quantities on the figure
Before you start calculating, mark up the figure. Finding right angles is key, because a right-angled triangle lets you connect two sides and an angle directly with sine, cosine or tangent; a triangle without a right angle needs the cosine formula or the sine formula. Right angles in a 3D figure usually come from the word 'perpendicular', from the corners of a cuboid, or from the question stating that a certain line is perpendicular to a certain plane.
Then write the lengths and angles given in the question beside the corresponding sides, and use different symbols to show whether it is a length or an angle that is wanted. This step looks superfluous, but it prevents confusion partway through: many mistakes are not from not knowing the formula but from putting the length of one side into the wrong triangle. Once the marking is done, two or three 'named' triangles have usually already emerged on the figure.
Step two: extract one plane triangle at a time
The way to extract a triangle is: choose three sides (or two sides and an angle), and draw them separately beside the figure as a plane triangle. When drawing, keep only these three sides and the relevant angle, and draw none of the other lines. The benefit is that the interference from the three-dimensional sense is removed, and you are facing an ordinary plane trigonometry question.
To judge whether a triangle is 'usable', ask three questions: how many of the three sides are known? Is there a right angle? Is the quantity wanted inside this triangle? If there is not enough known information, switch to another triangle, or first use another triangle to find the missing side. Here is a simple example: a cuboid has base length 8 and width 6, so the diagonal is the hypotenuse, and by Pythagoras' theorem 8² + 6² = 100, so the base diagonal is 10; if the height of the cuboid is 5, then the other right-angled triangle formed by the base diagonal and the height has hypotenuse 10² + 5² = 125, that is 5√5. This process is exactly extracting the base triangle first, then extracting another triangle, done in two steps.
- Measures, shape and spacePythagoras’ theorema2 + b2 = c2c is the hypotenuse of a right-angled triangle
- Measures, shape and spaceTrigonometric ratiossin θ = opphyp cos θ = adjhyp tan θ = oppadjFor right-angled triangles
- Measures, shape and spaceCosine formulac2 = a2 + b2 − 2ab cos CUse it with two sides and the angle between them, or three sides
- Measures, shape and spaceSine formulaasin A = bsin B = csin CUse it when you know two angles and a side
Step three: state the reason for each step
Once the triangle is extracted and the intermediate side calculated, put the result back into the 3D figure and extract the next triangle. When writing it up, it is advisable to write a reason beside each step, for example 'applying tangent in a right-angled triangle', 'found by Pythagoras' theorem', 'applying the cosine formula in that triangle'. This is not only for marks but also to check yourself: if you cannot write a reason, it usually means this triangle has not actually been proved to be right-angled, or the sides used do not lie in the same plane.
Public examinations such as those of the HKEAA value the process, but this article should not pronounce on how any examination marks; what can be said is that clearly writing the reasons lets the marker know which triangle you used, and makes it easier for you to check. If partway through you find the result is unreasonable, for example the hypotenuse turns out shorter than a leg, you should go back to the extracting step and check again rather than carry on calculating.
How to practise when revising
For everyday practice, you can focus on training the 'extracting the diagram' part rather than repeating lots of similar calculations. The method is: do only the first step of the question, that is mark the right angles and known quantities on the figure and list the order of triangles you intend to use, without completing the calculation. Doing a few of these exercises each day builds the judgement for choosing diagrams better than doing a whole paper in one go.
Another exercise is to draw the side view and plan view yourself first. For many 3D questions, as soon as you 'flatten out' a certain plane, the triangle appears naturally. Simple lines are enough when drawing; the point is that the positions of the side lengths and angles are correct. Afterwards, compare with the original figure in the question and check whether the two match; this comparison process is itself a good self-check. Primary school has not yet reached 3D trigonometry, but from P5 and P6 onwards, becoming fluent with the properties of plane figures and Pythagoras' theorem will help with extracting diagrams later.
Questions to ask when choosing a tutor
If you are considering tutoring, you can first distinguish several common forms: tutoring centres usually teach in small groups with a more uniform pace; the advantage of private tutoring and home tutoring is that they can target the individual's sticking points; online tutoring is more flexible in time and place. Prices vary widely between the three, generally calculated by lesson or by hour, roughly from about a hundred-odd dollars to several hundred dollars per hour; the actual fee depends on the form, year group and district, and you should ask clearly.
Whichever form you choose, you can first ask one specific question: when it comes to 3D trigonometry, will you require the student to extract the plane triangle first and state the reason? If the other party only emphasises memorising formulas or drilling answers, it may not solve the reader's difficulty. Bodies such as the Consumer Council have always reminded consumers to pay attention to contracts and fee terms; before paying, parents should understand the refund and make-up lesson arrangements. As for self-study, you can also use free online video courses to revisit topics, together with the extracting practice above.
3D trigonometry and further study: the foundation for university, applications and funding
3D trigonometry is an important topic in senior secondary maths, and also an introductory foundation for many university courses. Hong Kong has a wide variety of university degree courses, and some courses related to maths, engineering and social sciences require applicants to have a certain level of mathematical ability. If students plan to apply to university later, building a solid foundation in 3D trigonometry now will help with public examinations and also prepare them for future study.
In university admissions, besides public examination results, other factors may also be considered. On funding, Hong Kong has government grant and loan schemes to help eligible students pay tuition fees and living expenses. The eligibility and procedures for these schemes should be checked directly with the relevant departments or institutions; this article does not intend to set them out one by one.
In addition, some non-governmental organisations also provide scholarships or learning support, and students and parents can look out for community resources. As for choosing full-time courses, consider personal interest and ability rather than focusing only on whether individual subjects are easy or difficult.
Frequently asked questions
Must 3D trigonometry questions always use Pythagoras' theorem?
Not necessarily. Pythagoras' theorem only applies to right-angled triangles, to find the hypotenuse from the two legs or the other way round to find a side. If the triangle has no right angle, you need the cosine formula; when two angles and one side, or two sides and a non-included angle, are known, you may need the sine formula. The key to deciding is first to confirm whether there is a right angle in the triangle, then decide which formula to use.
Why do I not know how to continue after calculating the first step?
A common reason is not putting the result of the first step back into the 3D figure. After finding the intermediate side, mark its length on the figure, then look again for the triangle containing this side and the quantity wanted, and extract it to calculate separately. It is normal for a question to use two or three triangles; you need not expect to do it in one step.
How carefully should I draw when extracting a triangle?
Only draw the three sides you have chosen and the relevant angle; other lines can be left out. The figure need not be to scale, but the positions of known and unknown quantities should be marked clearly. If you draw it too complicatedly, you bring the interference of the 3D figure into the plane triangle instead.
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