Thinking [email protected] #5

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[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.


This is a very challenging trigonometry question from AJC Mid Years P1 Q10.

Using the formula for \text{sin}(A+B), prove that \text{sin}(3A) = 3 \text{sin} A - 4 \text{sin}^3 A .

(i) Hence show that \sum_{r=1}^n \dfrac{4}{3^r} \text{sin}^3 (3^r \theta) = \text{sin} (3 \theta) - \dfrac{1}{3^n} \text{sin}(3^{n+1} \theta).

(ii) Deduce the sum of the series \sum_{r=1}^n \text{sin}^2 (3^r \theta) \text{cos} (3^r \theta)

June Crash Course

June Crash Course

Chemistry, JC Chemistry, JC Mathematics, Mathematics

The team at The Culture SG has been really busy and we have a lot of things prepared to help you guys work for that A. First up! Crash course for June…

And we know it is a bit late to be announcing this on the site now, but we have really been caught up with preparing our students lately that we don’t have the time to properly update here. So here are the details for the Math Crash Course and the Chemistry Crash Course.

P.S. For SCIENCE students who wish to chiong in October, please take note that the H2 Chem/ Phy/ Bio Paper 4 (practical) is in October. So better start soon! Here are the details!

Click to view

For 3 hr lessons, they are priced at $105.

For 2 hr lessons, they are priced at $70.

Lessons will be held at:
Newton Apple Learning Hub
Blk 131, Jurong Gateway Road #03-263/265/267 Singapore 600131
Tel: +65 6567 3606

For math enquiries, you may contact Mr. Teng at +65 9815 6827.

For chem enquiries, you may contact Ms. Chan at +65 93494384.

For GP enquiries, you may contact Ms. Chen at +65 91899133.

Probability Question #4

Probability Question #4

JC Mathematics

A gambler bets on one of the integers from 1 to 6. Three fair dice are then rolled. If the gambler’s number appears k times (k = 1, 2, 3), he wins $ k. If his number fails to appear, he loses $1. Calculate the gambler’s expected winnings

Statistics related articles

JC Mathematics

Here is a compilation of all the Statistics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

1. What does unbiased in Math mean?
2. What is r-value?
3. Why is the r-value independent of translation and scaling? #1
4. Why is the r-value independent of translation and scaling? #2
5. Why can’t we extrapolate regression lines?
6. What is a regression line?
7. Simulation for Hypothesis Testing
8. Type I vs Type II Errors
9. A little about Central Limit Theorem
10. What is a good estimate?
11. Using symmetrical properties of Normal Curve to solve questions
12. Bayesian vs Frequentist Statistics

Integration related articles

JC Mathematics

Here is a compilation of all the Integration articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

1. Integrating Trigonometric Functions (1)
2. Integrating Trigonometric Functions (2)
3. Integrating Trigonometric Functions (3)
4. Integrating Trigonometric Functions (4)
5. Integrating Trigonometric Functions (5)
6. Leibniz’s Formula for \pi
7. Simpson’s Rule
8. Evaluating Integrals with Modulus