Trigonometric Functions, SPM Additional Mathematics Form 5

In your earlier years, trigonometry lived inside a right-angled triangle: $\sin$, $\cos$ and $\tan$ were ratios of sides, and every angle sat between $0^{\circ}$ and $90^{\circ}$. Trigonometric Functions takes those same three ratios and sets them free.

By placing an angle at the centre of a Cartesian plane, the chapter defines each ratio for any angle, obtuse, reflex, negative, or more than one full turn, and turns trigonometry from a triangle tool into a family of functions with graphs, patterns and identities of its own.

In SPM Additional Mathematics this is Chapter 6 of Form 5, and it belongs to the trigonometry strand of the course. It contains six content standards taken straight from the KSSM DSKP: positive and negative angles, trigonometric ratios of any angle, graphs of the sine, cosine and tangent functions, basic identities, the addition and double-angle formulae, and finally the application of trigonometric functions.

Each standard adds one layer, and by the end you can move fluently between a diagram, a graph, an identity and an equation.

One feature makes this chapter unusual and reassuring. Most of its headline formulae, the Pythagorean identities, the addition formulae for $\sin(A\pm B)$, $\cos(A\pm B)$ and $\tan(A\pm B)$, and the double-angle formulae, are printed for you in the list of formulae supplied in the SPM exam.

So the challenge here is rarely memory; it is knowing which formula to reach for and how to steer it toward the answer. The pieces you do have to hold in your head are smaller: the reciprocal definitions, the sign of each ratio in each quadrant, and the shape of each graph.

The chapter has two natural halves. The first half is about values and shapes: defining the ratios on the plane, using the CAST idea to fix signs, and sketching $y=a\sin bx+c$ and its cousins so you can read amplitude, period and the number of solutions from a picture.

The second half is about manipulation: proving identities, deploying the addition and double-angle formulae, and solving trigonometric equations over a stated range. Our teachers treat the chapter as a bridge, it rewards the neatness you built earlier and prepares the trigonometric derivatives and integrals you meet in the calculus chapters.

Because trigonometric functions reappear inside Differentiation, Integration and even Kinematics, time invested here pays off well beyond a single set of exam questions. A student who is comfortable finding every solution of $2\sin x=1$ in a given range, or rewriting $\cos 2x$ in three different forms, carries a quiet advantage into the rest of Form 5.

Trigonometric Functions is organised into six content standards. These codes and titles come directly from the DSKP, and recognising them helps you see exactly which skill a question is testing.

The six standards build in a deliberate order. Standards 6.1 and 6.2 fix what the ratios mean for any angle; 6.3 turns them into pictures; 6.4 and 6.5 give you the algebra of identities and compound angles; and 6.6 asks you to put everything to work in equations and problems.

If the sign work in 6.2 is shaky, both the graph sketches and the equation solving inherit the error, so make the quadrant signs automatic before you push on.

Angles can be positive or negative, and can go round more than once. Measured from the positive $x$-axis, an anticlockwise turn is a positive angle and a clockwise turn is a negative angle.

An angle such as $430^{\circ}$ is one full turn plus $70^{\circ}$, and $-40^{\circ}$ lands in the same place as $320^{\circ}$. Where the terminal arm finishes, which quadrant, is what decides the sign of each ratio.

On the plane, the ratios are defined by coordinates. Take a point $(x,y)$ on the terminal arm at distance $r=\sqrt{x^{2}+y^{2}}$ from the origin.

Then $\sin\theta=\frac{y}{r}$, $\cos\theta=\frac{x}{r}$ and $\tan\theta=\frac{y}{x}$. Because $x$ and $y$ can be negative while $r$ is always positive, this single definition extends the ratios to every angle and explains, cleanly, why a ratio is positive in some quadrants and negative in others.

The three reciprocal ratios are just flips. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent.

These definitions are not printed in the exam, so you must know them, and it helps to notice the mismatched pairing, cosecant goes with sine, and secant goes with cosine.

$$\operatorname{cosec}\theta=\dfrac{1}{\sin\theta},\qquad \sec\theta=\dfrac{1}{\cos\theta},\qquad \cot\theta=\dfrac{1}{\tan\theta}=\dfrac{\cos\theta}{\sin\theta}$$

The CAST rule fixes the sign in each quadrant. Reading the quadrants from the fourth round to the first, Cosine is positive in the fourth, All ratios are positive in the first, Sine is positive in the second, and Tangent is positive in the third.

To evaluate any angle, find its acute reference angle to the $x$-axis, take the ratio of that reference angle, then attach the sign the quadrant demands.

A handful of special angles have exact values worth knowing. The ratios of $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$ recur so often that recognising them saves time and sidesteps rounding.

Combined with the CAST rule, these exact values let you write down the ratio of any related angle in another quadrant, one whose reference angle is $60^{\circ}$, say, without reaching for the calculator at all. Negative angles follow a tidy symmetry too: reflecting an angle in the horizontal axis leaves cosine unchanged but reverses the sign of sine and tangent, so $\cos(-\theta)=\cos\theta$ while $\sin(-\theta)$ and $\tan(-\theta)$ simply change sign.

Each graph has an amplitude, a period and a vertical shift. For $y=a\sin bx+c$ and $y=a\cos bx+c$, the amplitude is $|a|$, the constant $c$ shifts the curve up or down, and $b$ squeezes the wave so more cycles fit in the same width.

The period, the width of one full cycle, follows from $b$. Tangent has no amplitude and repeats twice as often as sine and cosine.

$$\text{Period: } \dfrac{360^{\circ}}{b}\ (\sin,\ \cos), \qquad \dfrac{180^{\circ}}{b}\ (\tan)$$

Read the key features off each graph before you trust it. A good sketch shows where the curve crosses the axes, where it reaches its highest and lowest points, and, for tangent, where it shoots off to infinity because the function is undefined.

For $y=a\sin bx+c$ the maximum is $c+|a|$ and the minimum is $c-|a|$, so the whole wave sits in a band of height $2|a|$ centred on the line $y=c$. Marking those levels first keeps the shape honest and makes the later counting of solutions reliable.

Sketching turns an equation into a counting exercise. To find how many solutions an equation such as $a\sin bx+c=k$ has in a range, sketch the trigonometric curve and the straight line $y=k$ on the same axes and count the intersections.

This graphical method is exactly what standard 6.3.2 asks for, and it is far safer than guessing.

The three basic identities are the engine of every proof. Starting from $\sin^{2}A+\cos^{2}A=1$ and dividing through by $\cos^{2}A$ or $\sin^{2}A$ gives the other two.

All three are supplied in the exam, so proving an identity is about choosing the right one, not recalling it.

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

$$1+\tan^{2}A=\sec^{2}A \qquad 1+\cot^{2}A=\operatorname{cosec}^{2}A$$

Proving an identity follows a reliable strategy. Begin on the more complicated side and work steadily toward the simpler one, rather than rearranging both sides at once.

Rewriting everything in terms of sine and cosine often opens the path, and forming a common denominator will frequently expose a Pythagorean identity ready to be substituted. When the two sides finally agree, say so, the marks reward one clean, one-directional chain of working, not a scramble that meets in the middle.

The addition formulae combine two angles into one expression. They let you expand $\sin(A\pm B)$, $\cos(A\pm B)$ and $\tan(A\pm B)$, and they are the source of the double-angle results.

Notice the sign flips: cosine reverses the sign in the middle, so $\cos(A+B)$ uses a minus. All three are supplied in the exam.

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

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

$$\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$

The double-angle formulae are the addition formulae with $B=A$. Setting the two angles equal gives $\sin 2A$, $\cos 2A$ and $\tan 2A$; the cosine version has three equivalent forms, and choosing the one that matches your goal is a real exam skill, for example, $\cos 2A=1-2\sin^{2}A$ is exactly what you want when only sines appear.

These too are supplied in the exam.

$$\sin 2A=2\sin A\cos A \qquad \tan 2A=\dfrac{2\tan A}{1-\tan^{2}A}$$

$$\cos 2A=\cos^{2}A-\sin^{2}A=2\cos^{2}A-1=1-2\sin^{2}A$$

Solving an equation means finding every solution in the range. Reduce the equation to a single ratio, find the basic (reference) angle from the positive value, then use CAST to place the answer in each quadrant the sign allows, adding or subtracting $360^{\circ}$ (or $180^{\circ}$ for tangent) until the whole stated range is filled.

The range in the question, for example $0^{\circ}\le x\le 360^{\circ}$, is an instruction about how many answers to give, not a detail to skim past.

Trigonometric Functions can appear in either written paper. Paper 1 (3472/1) lasts 2 hours and carries 80 marks: Section A has 12 questions worth 64 marks that you answer all of, and Section B has 3 questions worth 16 marks from which you answer 2.

Paper 2 (3472/2) lasts 2 hours 30 minutes and carries 100 marks across Section A (7 questions, 50 marks, answer all), Section B (4 questions, 30 marks, answer 3) and Section C (4 questions, 20 marks, answer 2).

Across both papers the items are limited-response subjective and structured questions, they are marked with analytic scoring, and you sit them using a non-programmable scientific calculator. The chapter suits the longer structured items well, because a single stem can chain several parts, prove an identity, then use it to simplify, then solve the resulting equation over a range, with marks accumulating step by step.

Sketching questions and identity proofs are natural fits for the extended sections. We do not predict how many marks any single chapter carries, since that varies from year to year.

Because the scoring is analytic, your method earns marks even when a final value is slightly out, so always show the identity or formula you started from, each line of the manipulation, and the reference angle you used before listing the solutions. Our lessons are taught in English while SPM papers are set bilingually in Malay and English, so you will meet the key terms, identity, amplitude, period, quadrant, in both languages and can recognise what a question wants however it is phrased.

Most marks lost in Trigonometric Functions come from a handful of recurring habits, and each one is easy to fix once you can name it. Read these before every practice set until avoiding them is automatic.

• Giving only one solution when the range holds more. An equation such as $\sin x=0.5$ has two answers in $0^{\circ}\le x\le 360^{\circ}$, not one. Find the basic angle, then use CAST to place a solution in every quadrant where the sign fits before you stop.
• The calculator in the wrong mode. If the range is in degrees but the calculator is in radian mode (or the reverse), every value comes out wrong. Match the mode to the range at the very start of the question and glance at the display indicator to confirm it.
• Misremembering the quadrant signs. Cosine is negative in the second quadrant and tangent is positive in the third, getting CAST backwards flips the sign of the answer. Draw the CAST diagram in the margin rather than trusting memory under pressure.
• Dividing an equation by $\sin x$ or $\cos x$. Cancelling a trigonometric factor throws away the solutions where that factor is zero. Move everything to one side and factorise instead, then solve each factor equal to zero.
• The sign slip in $\cos(A+B)$. The cosine addition formula reverses the sign in the middle: $\cos(A+B)=\cos A\cos B-\sin A\sin B$. Writing a plus there is one of the most common proof errors, copy the $\mp$ from the formula list carefully.
• Reading a graph's period straight off $b$. In $y=a\sin bx+c$ the period is $\frac{360^{\circ}}{b}$, not $b$ itself, and the amplitude is $|a|$, not $a+c$. Label amplitude, period and shift separately before you sketch.

None of these is about ability, each is a small habit you can tick off during practice. Students who score well in this chapter are simply the careful ones: mode matched to the range, CAST drawn out, factors kept rather than cancelled, and the range filled to the end.

Treat every slip as feedback rather than failure, and your accuracy climbs quickly.

Trigonometric Functions rewards a steady routine, settle the signs, learn the shapes, then practise the algebra. Work through the steps below, then use the resources that follow to revise and test yourself.

Keep your sessions short and frequent rather than one long push. The two skills that unlock everything else, placing an angle in the right quadrant, and finding every solution in a range, are built by repetition, so a few mixed questions each day beat an occasional marathon.

When signs and graphs feel automatic, move on to identity proofs and equations; when they do not, slow down and drill the CAST diagram until it is instant.

1. 1

   Make the quadrant signs automatic

   Practise the CAST rule and reference angles until you can state the sign of $\sin$, $\cos$ and $\tan$ in any quadrant without hesitating, and evaluate any-angle ratios on sight.

2. 2

   Learn the three graph shapes

   Sketch $y=a\sin bx+c$, $y=a\cos bx+c$ and $y=a\tan bx+c$ from their amplitude, period and shift, then read the number of solutions by adding the line $y=k$.

3. 3

   Drill identity proofs

   Prove identities by working one side at a time, reaching first for $\sin^{2}A+\cos^{2}A=1$ and its two partners, and only then for the addition and double-angle formulae.

4. 4

   Master the compound-angle formulae

   Expand with the addition formulae and rewrite $\sin 2A$, $\cos 2A$ and $\tan 2A$ fluently, choosing the cosine form that matches whatever the question already contains.

5. 5

   Solve equations over a range under time

   Reduce to one ratio, find the basic angle, use CAST to list every solution in the stated range, then attempt full Paper 2-style questions with a clock running and review the worked examples afterwards.

• Trigonometric Functions, Revision Notes
• Trigonometric Functions, Common Mistakes
• Trigonometric Functions, Practice Questions
• Trigonometric Functions, Worked Examples (Easy)
• Trigonometric Functions, Worked Examples (Medium)
• Trigonometric Functions, Worked Examples (KBAT)
• Trigonometric Functions, Paper 2 Answering Guide
• Trigonometric Functions, Key Terms