When you see a trigonometric identity, the hardest part can be choosing the first step. Start by looking at what changes between the two sides. Do the functions change, do the angles change, or is there a familiar algebra pattern?
The O-Level Additional Mathematics syllabus covers trigonometric identities and proofs, equations, graphs, and rewriting expressions in R-form. Here are four strategies we teach Secondary 3 and 4 students at Jimmy Tuition. SEAB O-Level Additional Mathematics syllabus (4049).
1. Rewrite unfamiliar ratios in sine and cosine
When a proof mixes tan, cot, sec and cosec, try rewriting the more complicated side in sin and cos. This often makes it easier to combine fractions or spot a common factor.
- tan x = (sin x) / (cos x)
- cot x = (cos x) / (sin x)
- sec x = 1 / (cos x)
- cosec x = 1 / (sin x)
Each formula applies where its denominator is nonzero. Start with one side of the identity and simplify it until it matches the other. For example, sec x − cos x = (1 − cos2 x) / (cos x) = (sin2 x) / (cos x) = sin x tan x, for cos x ≠0.
2. Let the angles guide your choice of formula
If you see A + B or A − B, look for an addition or subtraction identity. If you need to connect 2A with A, try a double-angle identity.
Cosine has three useful double-angle forms. Choose the one that matches the terms you want to keep or cancel.
- cos(2A) = cos2 A − sin2 A
- cos(2A) = 2cos2 A − 1. This makes 1 + cos(2A) simplify to 2cos2 A.
- cos(2A) = 1 − 2sin2 A. This makes cos(2A) − 1 simplify to −2sin2 A.
3. Look for familiar algebra patterns
- Difference of squares. 1 − sin2 x = (1 − sin x)(1 + sin x). You can also replace 1 − sin2 x with cos2 x if that gets you closer to the other side.
- Perfect square. sin2 x + 2sin x cos x + cos2 x = (sin x + cos x)2.
- Common factor. sin x cos x + sin x = sin x(cos x + 1). Spotting the factor can turn a long equation into two shorter ones.
4. Use the quadrant rule to solve equations
An identity holds wherever both sides are defined. A trigonometric equation asks for the angles that satisfy it in a specified interval. Your calculator may give just one of those angles.
- Find the reference angle. For a solution inside a quadrant, use the magnitude of the isolated ratio to find the acute angle. Check axis angles directly when the ratio gives one, such as sin θ = 0 or cos θ = 1.
- Use ASTC to choose quadrants. In quadrant I, all three ratios are positive. Sine is positive in II, tangent in III and cosine in IV. Use the sign of the original ratio.
- Check the whole interval. Match the calculator mode to the question. Include endpoints only if the interval includes them and they satisfy the original equation. For wider intervals, use the period to find any further solutions.
For example, if sin θ = −1/2 and 0° ≤ θ < 360°, the reference angle is 30°. Sine is negative in quadrants III and IV, so θ = 210° or 330°.
Three common exam mistakes
- Using the wrong calculator mode. Check whether the question uses degrees or radians before finding an inverse trig value.
- Dividing by a term that can be zero. In 2sin x cos x = sin x, dividing by sin x can lose the solutions where sin x = 0. Factor instead to get sin x(2cos x − 1) = 0 and check both cases.
- Keeping extra solutions after squaring. Squaring can introduce values that fail the original equation. Substitute each candidate back to check.
Practise A-Math with Jimmy Tuition
We help Secondary 3 and 4 students build a step-by-step approach to proofs and equations, whether they need to revisit the basics or practise under exam conditions.
- In-person classes. Our centre is a three-minute sheltered walk from Admiralty MRT, near schools such as Admiralty Secondary, Woodlands Ring Secondary, Evergreen Secondary and Woodgrove Secondary.
- Live online lessons. Join interactive lessons with a digital whiteboard and contact Mr. Jimmy on WhatsApp for individual help outside class hours.
For more on class sizes, makeup lessons and materials, see our Frequently Asked Questions.
