Trigonometry Formula For O levels A Maths
| Trigonometry Formula |
|---|
| sin2 A + cos2 A = 1 |
| sec2A + 1 = tan2 A |
| cosec2A = 1 + cot2A |
| sin(A±B) = sin A cos B ± cos A sin B |
| cos(A±B) = cos A cos B ∓ sin A sin B |
| tan(A±B) = tan A ± tan B ⁄1∓tan A tan B |
| sin2A = 2 sinA cosA |
| cos2A = cos2A - sin2A = 2cos2A - 1 = 1 -2sin2A |
| tan2A = 2tan A ⁄1-tan2A |
Proving trigonometries identities is a type of question that is struggled by many students. The common errors can stem from having serious algebraic misconceptions that they carry forward from lower secondary or they simply do not know the proper techniques to prove. More often than not, students will basically apply on whatever formulas they can see before their eyes, taking a big detour to get to the answer (if they manage to get to their destination). We should prescribe the right medicine for a particular illness and cherry pick the formulas by observation. All these will come with consistent practices and good observation skills. Students who are particularly good at proving may not have high intelligence, one thing we can be sure is that they have their definite share of hard work put in and had reaped their rewards. Another point that I would like to point out is to MEMORISE ALL trigonometry formulas and know them at the back of their hands for efficiency of the chapters. We do not have the luxury of staring at the formula list in examinations and take our own sweet time. Let us move on to these 2 questions from Chung Cheng High and Nanyang Girls High.
Rules:
- Start from the side that contains \(A \pm B\) (instead of \(AB\) or \(\frac{A}{B}\) ) or the side which looks more complicated.
- Change all trigonometry to SINE or COSINE (these 2 are the basic trigo)
- Factorise and/or Make into same denominator where necessary.
- Application of trigonometry formulas
Note that Rules 2, 3 and 4 are not in running orders and are still dependent on observation skills of the trigonometry present on both sides. However, these are tried and tested rules that work well most of the times based on years of experience. The exceptions I wont cover here as these are special cases. All my students can work pretty fast using the rules.
Chung Cheng High (Yishun) Sec 4 Prelim 2017 P2/Q12
Example 1: Prove the identity \(\left(\frac{1}{\sin \theta} – \frac{1}{\tan \theta} \right)^2 = \frac{1-\cos\theta}{1+\cos\theta}\).
LHS
\(=\left(\frac{1}{\sin \theta} – \frac{1}{\tan \theta} \right)^2\)
\(=\left(\frac{1}{\sin \theta} – \frac{1}{\left(\frac{\sin \theta}{\cos \theta} \right)} \right)^2 \)
\(=\left(\frac{1}{\sin \theta} – \frac{\cos \theta}{\sin \theta} \right)^2\)
\(=\left(\frac{1-\cos\theta}{\sin \theta} \right)^2\)
\(=\frac{(1-\cos\theta)^2}{\sin^2\theta}\)
\(=\frac{(1-\cos\theta)^2}{1-\cos^2\theta}\)
\(=\frac{(1-\cos\theta)^2}{1-\cos^2\theta}\)
\(=\frac{(1-\cos\theta)^2}{(1+\cos\theta)(1-\cos\theta)}\)
\(=\frac{1-\cos\theta}{1+\cos\theta}\)
=RHS
Voilà! We achieved the proof by making use of the standard rules.
Nanyang Girls High 2017 Sec 4 MYE P2/Q10
Example 2: Show that \(\frac{2\cos^2y+\cot y}{1+\text{cosec }2y}=2\cos^2y\).
LHS
\(=\frac{2\cos^2y+\cot y}{1+\text{cosec }2y}\)
\(=\frac{\left(2\cos^2y + \frac{\cos y}{\sin y}\right)}{1 + \frac{1}{\sin 2y}}\)
\(=\frac{\left(\frac{2\cos^2y\sin y+\cos y}{\sin y}\right)}{\left(\frac{\sin 2y+1}{\sin 2y}\right)}\)
\(=\frac{\cos y(2\sin y \cos y + 1)}{\sin y} \times \frac{\sin 2y}{\sin 2y+1}\)
\(=\frac{\cos y(\sin 2y + 1)}{\sin y} \times \frac{2\sin y \cos y}{\sin 2y + 1}\)
\(=2\cos^2y\)
=RHS
