Trigonometric functions and identities explained

You arrive at this chapter knowing three trigonometric ratios: sine, cosine, and tangent. Add Math extends them in two ways.

First, it introduces the three reciprocal functions, cosecant, secant, and cotangent, which are simply one divided by the ones you know. Second, it stops restricting angles to the acute angles of a right-angled triangle and lets $\theta$ be any angle at all, positive or negative, by placing it on a set of axes.

The three reciprocals are worth stating plainly, because the pairing is not the obvious one:

$$\csc\theta=\frac{1}{\sin\theta}\qquad \sec\theta=\frac{1}{\cos\theta}\qquad \cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}$$

Once angles can exceed $90^\circ$, the functions can be positive or negative depending on where the angle lands. The plane splits into four quadrants, and in each one a different set of functions is positive.

The standard memory aid is the phrase 'All, Sin, Tan, Cos' read anticlockwise from the first quadrant:

• First quadrant ($0^\circ$ to $90^\circ$): All three of sin, cos and tan are positive.
• Second quadrant ($90^\circ$ to $180^\circ$): only Sin (and its reciprocal cosec) is positive.
• Third quadrant ($180^\circ$ to $270^\circ$): only Tan (and its reciprocal cot) is positive.
• Fourth quadrant ($270^\circ$ to $360^\circ$): only Cos (and its reciprocal sec) is positive.

This is exactly what you need when a question says something like '$\sin\theta=0.6$ and $\theta$ is obtuse'. Obtuse puts $\theta$ in the second quadrant, where cosine is negative, so once you find $\cos\theta$ has size $0.8$, the sign rule tells you it must be $-0.8$.

Getting the sign right is worth as many marks as getting the number right.

An identity is an equation true for every angle, and the whole point of learning them is to rewrite a messy expression into a simpler one. The foundation is the Pythagorean identity, which comes straight from the theorem applied on the unit circle:

$$\sin^{2}\theta+\cos^{2}\theta=1$$

Divide this one identity through by $\cos^{2}\theta$, and then by $\sin^{2}\theta$, and you get its two partners for free:

$$1+\tan^{2}\theta=\sec^{2}\theta\qquad 1+\cot^{2}\theta=\csc^{2}\theta$$

Beyond these come the compound-angle (addition) formulas and the double-angle formulas, which appear on the formula list you are given in the exam. The compound-angle forms are:

$$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\qquad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$

Set $B=A$ in these and you recover the double-angle formulas, which is worth doing once yourself so you never fear forgetting them: $\sin 2A=2\sin A\cos A$ and $\cos 2A=\cos^{2}A-\sin^{2}A$. That last one has two handy rewrites using $\sin^{2}A+\cos^{2}A=1$: $\cos 2A=1-2\sin^{2}A=2\cos^{2}A-1$.

Choosing the version that matches the rest of the question is a real exam skill.

Two question types dominate this chapter. The first asks you to prove an identity, to show the left-hand side equals the right.

The reliable method is to start with the more complicated side and rewrite it, step by step, until it matches the other side. Convert everything to sines and cosines, use $\sin^{2}\theta+\cos^{2}\theta=1$ to simplify, and never move terms across the equals sign as if solving, a proof works down one side until it becomes the other.

The second type asks you to solve a trigonometric equation within a stated range, such as $0^\circ\le\theta\le 360^\circ$. Here the sign rules return with force: an equation like $\sin\theta=0.5$ has more than one solution in that range (one in the first quadrant, one in the second), and the marks depend on finding all of them.

Sketch the graph or the quadrant diagram, find the basic angle, then use the quadrants to list every valid solution in the range.

The recurring ways marks slip away in this chapter are worth naming:

• Wrong reciprocal pairing, writing $\sec\theta=\frac{1}{\sin\theta}$. Sec pairs with cos; cosec pairs with sin.
• Sign errors from ignoring the quadrant. When you take a square root to find, say, $\cos\theta$, the algebra gives the size; the quadrant gives the sign. You need both.
• Treating an identity proof like an equation and shuffling terms across the equals sign. Work down one side only.
• Losing solutions in a range, and mixing up degree and radian mode on the calculator when the range is given in radians.

Because SPM Add Math is marked analytically, clean line-by-line working protects you here more than in almost any other chapter. A proof that stalls two lines from the end still earns marks for every correct step; an equation where you find three of four solutions still scores the three.

Write the identity you are using, write the substitution, and show each simplification, the method marks add up.

Trigonometric identities are the chapter where students most often say 'I understand it when I watch, but I freeze on a blank page'. That gap between recognising and generating is exactly what a live teacher closes: watching you attempt a proof, they can see the moment you hesitate over which identity to reach for, and coach the decision until choosing your first move feels natural.

Our teachers are experienced, and lessons are online and taught in English, which also helps with the English terms, while SPM papers are set in both Bahasa Melayu and English.

If you would like to try a session, the first lesson is a one-hour paid class at the teacher's rate, from RM50 an hour, depending on the teacher's experience, quoted on WhatsApp. We start from where you are, whether that is the reciprocal functions or proving identities, and build outward at your own pace, without pressure.