Trigo Formulae

The following formulae will be useful when integrating Trigonometric functions. Taken from the MF15 formula sheet for JC.

Addition Formulae

$\begin{aligned} \sin(A\pm B)&\equiv\sin A\cos B\pm\cos A\sin B\\ \cos(A\pm B)&\equiv\cos A\cos B\mp\sin A\sin B\\ \tan(A\pm B)&\equiv\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\\ \end{aligned}$

Double Angle Formulae

$\begin{aligned} \sin 2A &\equiv 2\sin A\cos A\\ \cos 2A\equiv\cos^2 A-\sin^2 A&\equiv 2\cos^2 A-1\equiv 1-2\sin^2 A\\ \tan 2A&\equiv\frac{2\tan A}{1-\tan^2 A} \end{aligned}$

Remark: The second identity is useful for integrating $\sin^2 x$ and $\cos^2 x$ .

Factor Formulae

$\begin{aligned} \sin P+\sin Q&\equiv 2\sin\frac 12(P+Q)\cos\frac 12(P-Q)\\ \sin P-\sin Q&\equiv 2\cos\frac 12(P+Q)\sin\frac 12(P-Q)\\ \cos P+\cos Q&\equiv 2\cos\frac 12(P+Q)\cos\frac 12(P-Q)\\ \cos P-\cos Q&\equiv -2\sin\frac 12(P+Q)\sin\frac 12(P-Q) \end{aligned}$

Remark: The factor formulae are useful for integrating $\sin nx\cos mx$ , $\sin nx\sin mx$ , etc.

Topics coming out for A Maths Paper 2

Recently, the A Maths Paper 2 just finished (today), and the A Maths Paper 2 is hot on its heels, coming tomorrow!

Usually, topics tested in Paper 1 will most likely not come out again in Paper 2, so students doing their last minute revision can use this fact to focus their revision.

Topics that came out in Paper 1:

Binomial Theorem
Trigonometry (Addition Formula, find exact value)
Rate of change
Partial Fractions
Linear Law
Prove Trigonometry
Coordinate Geometry
Integrate & Differentiate Trigonometric Functions
Discriminant (b^2-4ac)
Stationary Points
Tangent/Normal
Quadratic/Modulus

Topics likely to come out in Paper 2:

Indices/Surds
Polynomials
Exponential / Logarithmic Equations
R-formula
Sketching of Trigonometric Graphs
Circles
Proofs in plane geometry
Integration as the reverse of differentiation
Area under curve
Kinematics

All the best for those who are taking the A Maths Paper 2 exam tomorrow. 🙂

Challenging O Level Trigonometry Question (A Maths)

A reader of our Mathtuition88.com blog asked the following Maths question:

Given that $\sin x+\sin y=a$ and $\cos x+\cos y=a$ , where $a\neq 0$ , express $\sin x+\cos x$ in terms of $a$ .

This is a rather challenging question, since there are many options to start. Which formula(s) should we use? Factor formula? R-formula? Give it a try first if you want to have a challenge.

Solution:

It turns out we can write:

$\sin y=a-\sin x$

$\cos y=a-\cos x$

Then, use $\sin^2 y+\cos ^2 y=1$

$(a-\sin x)^2+(a-\cos x)^2=1$

Expanding,

$a^2-2a\sin x+\sin^2 x+a^2-2a\cos x+\cos^2 x=1$

Rearranging,

$2a^2-2a(\sin x+\cos x)+1=1$

$2a(a-(\sin x+\cos x))=0$

Since $a\neq 0$ , we have $a-(\sin x+\cos x)=0$ .

Thus, $\boxed{\sin x+\cos x=a}$ .

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Schaum’s Outlines–Problem Solved.

Mr Bean Calculus and Trigonometry Math Exam

For O Levels, we need to be proficient in both Calculus (Differentiation and Integration), AND Trigonometry! Students need to revise early to avoid being like Mr Bean!

sin(180-x)

Trigonometry Identities

Negative angles:

$\sin (-x)=-\sin x$
$\boxed{\cos (-x)=\cos x}$ (Still Positive!)
$\tan (-x)=-\tan x$

Reason: $-x$ is in the “C” quadrant so Cosine is still positive.

Supplementary angles:

$\boxed{\sin (180^\circ -x)=\sin x}$ (Still positive!)
$\cos (180^\circ -x)=-\cos x$
$\tan (180^\circ -x)=-\tan x$

Reason: $180^\circ -x$ is in the “S” quadrant so Sine is still positive.

Complementary angles:

$\sin (90^\circ -x)=\cos x$
$\cos (90^\circ -x)=\sin x$
$\tan (90^\circ -x)=\cot x$

Reason:

$\sin (90^\circ -x)=\cos x=a/c$
$\cos (90^\circ -x)=\sin x=b/c$
$\tan (90^\circ -x)=\cot x=a/b$

How to Remember Trigonometric Special Angles easily using Calculator

How to Remember Special Angles easily using Calculator

Step 1: Type the expression into calculator, eg. $\sin (\frac{\pi}{3})$ (in radian mode, for this case)

Step 2: You will get 0.8660254038. Square the Answer. (Ans^2)

Step 3: You will get 3/4. That means $\sin (\frac{\pi}{3})=\frac{\sqrt{3}}{2}$

Challenging Trigonometry Question (ACS(I) Sec 3)

Solution:

(a) $\tan (\alpha + \beta )=\frac{10}{x}$

Using the formula,

$\displaystyle \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta}=\frac{10}{x}$

$\displaystyle \frac{\frac{5}{x}+\tan \beta}{1-(\frac{5}{x})\tan \beta}=\frac{10}{x}$

Cross-multiply,

$5+x\tan\beta =10-\frac{50}{x}\tan \beta$

$(x+\frac{50}{x})\tan\beta =5$

$\displaystyle \tan \beta =\frac{5}{x+\frac{50}{x}}=\frac{5x}{x^2+50}$

(b) The trick here is to break up $2\alpha +\beta$ into $\alpha + (\alpha +\beta )$

$\displaystyle \begin{array}{rcl} \tan (2\alpha +\beta )&=& \tan (\alpha + (\alpha + \beta ))\\ &=& \frac{\tan \alpha + \tan (\alpha + \beta )}{1-\tan \alpha \tan (\alpha + \beta )}\\ &=& \frac{ \frac{5}{x}+\frac{10}{x} }{ 1-(\frac{5}{x})(\frac{10}{x}) }\\ &=& \frac{\frac{15}{x}}{\frac{x^2-50}{x^2}}\\ &=& \frac{15x}{x^2-50} \end{array}$

Range:

Since $2\alpha +\beta$ is acute (1st quadrant), $\tan (2\alpha +\beta )$ is positive.

$x^2-50 >0$

$x>\sqrt{50}$

积少成多: How can doing at least one Maths question per day help you improve! (Maths Tuition Revision Strategy)

We all know the saying “an apple a day keeps the doctor away“. Many essential activities, like eating, exercising, sleeping, needs to be done on a daily basis.

Mathematics is no different!

Here is a surprising fact of how much students can achieve if they do at least one Maths question per day. (the question must be substantial and worth at least 5 marks)

This study plan is based on the concept of 积少成多, or “Many little things add up“. Also, this method prevents students from getting rusty in older topics, or totally forgetting the earlier topics. Also, this method makes use of the fact that the human brain learns during sleep, so if you do mathematics everyday, you are letting your brain learn during sleep everyday.

Let’s take the example of Additional Mathematics.

Exam is on 24/25 October 2013.

Let’s say the student starts the “One Question per day” Strategy on 20 May 2013

Days till exam: 157 days (22 weeks or 5 months, 4 days)

So, 157 days = 157 questions (or more!)

Each paper in Ten Year Series has around 25 questions (Paper 1 & Paper 2), so 157 questions translates to more than 6 years worth of practice papers! And all that is achieved by just doing at least one Maths question per day!

A sample daily revision plan can look like this. (I create a customized revision plan for each of my students, based on their weaknesses).

Suggested daily revision (Additional Mathematics):

	Topic
Monday	Algebra
Tuesday	Geometry and Trigonometry
Wednesday	Calculus
Thursday	Algebra
Friday	Geometry and Trigonometry
Saturday	Calculus
Sunday	Geometry and Trigonometry

(Calculus means anything that involves differentiation, integration)

(Geometry and Trigonometry means anything that involves diagrams, sin, cos, tan, etc. )

(Algebra is everything else, eg. Polynomials, Indices, Partial Fractions)

By following this method, using a TYS, the student can cover all topics, up to 6 years worth of papers!

Usually, students may accumulate a lot of questions if they are stuck. This is where a tutor comes in. The tutor can go through all the questions during the tuition time. This method makes full use of the tuition time, and is highly efficient.

Personally, I used this method of studying and found it very effective. This method is suitable for disciplined students who are aiming to improve, whether from fail to pass or from B/C to A. The earlier you start the better, for this strategy. For students really aiming for A, you can modify this strategy to do at least 2 to 3 Maths questions per day. From experience, my best students practice Maths everyday. Practicing Ten Year Series (TYS) is the best, as everyone knows that school prelims/exams often copy TYS questions exactly, or just modify them a bit.

The role of the parent is to remind the child to practice maths everyday. From experience, my best students usually have proactive parents who pay close attention to their child’s revision, and play an active role in their child’s education.

This study strategy is very flexible, you can modify it based on your own situation. But the most important thing is, practice Maths everyday! (For Maths, practicing is twice as important as studying notes.) And fully understand each question you practice, not just memorizing the answer. Also, doing a TYS question twice (or more) is perfectly acceptable, it helps to reinforce your technique for answering that question.

If you truly follow this strategy, and practice Maths everyday, you will definitely improve!

Hardwork $\times$ 100% = Success! (^_^)

“There is no substitute for hard work.” – Thomas Edison

A Maths Tuition: Trigonometry Formulas

Many students find Trigonometry in A Maths challenging.

This is a list of Trigonometry Formulas that I compiled for A Maths. Students in my A Maths tuition class will get a copy of this, neatly formatted into one A4 size page for easy viewing.

A Maths: Trigonometry Formulas

$\mathit{cosec}x=\frac{1}{\sin x}$ ; $\mathit{sec}x=\frac{1}{\cos x}$