The most-tested knowledge point is in trigonometry
If you are preparing for undergraduate study in China, CSCA math is a hurdle you cannot avoid, and trigonometry makes up a substantial part of it.
The official syllabus lists trigonometry under "basic elementary functions" in the "Functions" module. In other words, trigonometry is not a separate block of knowledge but is tested alongside functions — so when you study it, treat it as functions.
We counted 7 past papers containing 333 questions in total. The method was simple: each question was classified by its main testing point into a single knowledge point. The result is that the 9 trigonometry-related knowledge points account for 71 questions, about 21%, which means roughly one question in five is trigonometry.
More noteworthy is that a single knowledge point, "definition and relations of trigonometric functions", accounts for 36 questions. That is the most heavily tested knowledge point among all 333, and it appeared in all 7 papers. For international students, these are marks you must secure.
Question counts for the nine knowledge points
The list below runs from most to fewest questions, with the number of papers out of 7 in which the knowledge point appeared in parentheses. Once this order is clear to you, you will know where to spend your time first.
- Definition and relations of trigonometric functions: 36 questions, tested in all 7 papers
- Graphs and properties of sine and cosine functions: 10 questions, tested in all 7 papers
- Induction formulas for trigonometric functions: 7 questions, tested in 6 papers
- Double-angle sine, cosine and tangent formulas: 7 questions, tested in 6 papers
- Graphs and properties of tangent and cotangent functions: 5 questions, tested in 4 papers
- Half-angle sine, cosine and tangent formulas: 5 questions, tested in 3 papers
- Product-to-sum and sum-to-product formulas: 1 question, tested in 1 paper
- Sine, cosine and tangent formulas for the sum and difference of two angles: 0 questions
- Definition of angles and radians: 0 questions
Two points need explaining.
First, each question is assigned to only one main testing point, so the numbers here add up to exactly 71, with no double counting.
Second, "sum and difference formulas" and "radians" come out at 0, but that does not mean you can skip them: questions that use them were classified under other knowledge points. The most typical example is the double-angle formula — cos 2α = cos²α − sin²α is derived from the sum formula. If you do not understand the sum formula, you can only memorize the double-angle formula, and it is easy to slip up once a question is changed. So you still need to study both items; you simply do not need to drill as many questions on them as on "definition and relations".
What You Must Memorize
This section is a ready-made list you can memorize directly. The formulas are written in plain text, so you can copy them onto paper and review them once a day.
Values of special angles
- sin30° = 1/2, cos30° = √3/2
- sin45° = cos45° = √2/2
- sin60° = √3/2, cos60° = 1/2
- tan45° = 1
Calculators are not allowed in the exam, so you must be able to recall these values instantly rather than work them out in the exam hall.
Same-angle relationships
- sin²α + cos²α = 1
- tanα = sinα / cosα
These are the core of “definition and relations”.
Radian conversion
- 180° = π radians
Induction formulas
The mnemonic is “odd changes, even stays the same; the sign depends on the quadrant.” Here are two examples:
- sin(π − α) = sinα
- cos(π − α) = −cosα
Use these two examples first to understand what the mnemonic means, then extend it to other angles.
Double-angle formulas
- sin2α = 2sinα·cosα
- cos2α = cos²α − sin²α = 1 − 2sin²α = 2cos²α − 1
cos2α has three forms. In the exam, choose the one that best suits the conditions given in the question.
Graphs and properties
- The sine and cosine functions have a period of 2π and a range of [−1, 1].
- The minimum positive period of y = A·sin(ωx + φ) is 2π/|ω|.
How to Practice
Memorizing is not the same as being able to solve problems, so practice in order.
- First, practice “definition and relations” until they become automatic. Given sinα and the quadrant the angle is in, find cosα and tanα — this is the core skill of this knowledge point. Practice until you can start writing the moment you see the question, without first thinking about the formula.
- Then practice graphs and properties: period, maximum and minimum values, and intervals of monotonicity. There are not many questions in this part (10 questions), but it appears in all 7 papers, so it is worth securing.
- Finally, practice formula transformations: induction formulas, double-angle formulas and half-angle formulas. Half-angle formulas have appeared in only 3 papers, so you can leave them until last — but do not skip them completely.
There is also a habit to build early: calculators are not allowed in the exam, so do not rely on one to remember the values of special angles. Rely on repeated writing and repeated recitation until they become automatic.
Next Steps
After reading this, do not rush to memorize all the formulas. It is recommended that you first take a 15-minute diagnostic to see which part of trigonometry needs the most work: definition and relations, graphs and properties, or formula transformations.
After the diagnostic, go to the Learning Center and practice knowledge point by knowledge point: first “definition and relations,” then graphs and properties, and finally formula transformations.