Trigonometry for IGCSE Mathematics - Additional: Methods and Worked Examples

The maths behind every wave, wheel, and orbit

Picture a lighthouse beam sweeping across the sea at night. That beam traces a circle, and the height of its light on a distant cliff face rises and falls in a smooth, repeating pattern. That pattern is trigonometry in action. If you've ever watched a Ferris wheel or listened to music through headphones, you've experienced the same idea: something rotating creates a wave, and waves are described by sine, cosine, and their relatives.

In IGCSE Additional Mathematics, trigonometry goes well beyond the SOHCAHTOA basics you met in 0580. You'll work with six trigonometric functions instead of three, prove identities that connect them, solve equations that have multiple solutions, and sketch graphs that model real oscillating behaviour. It sounds like a lot, but every piece builds on the last, and once you see how the six functions are just different views of the same rotating point, the whole topic clicks into place.

Meeting the full family: six trigonometric functions

You already know sine, cosine, and tangent. The three newcomers are simply their reciprocals.

Think of it like a family portrait. Sine and cosecant are partners, cosine and secant are partners, tangent and cotangent are partners. Whenever you see cosec, your brain should immediately whisper "that's 1 over sin." Building that reflex saves time under exam pressure.

Angles of any magnitude and the CAST diagram

In the right-angle triangle world, angles only go from 0 to 90 degrees. But Additional Maths needs you to handle any angle: 150 degrees, 240 degrees, negative angles, whatever the question throws at you. The key tool here is the CAST diagram.

Imagine the coordinate axes dividing a circle into four quadrants. Starting from the positive x-axis and going anticlockwise:

• First quadrant (0 to 90 degrees): All six functions are positive
• Second quadrant (90 to 180 degrees): Only Sine (and cosecant) are positive
• Third quadrant (180 to 270 degrees): Only Tangent (and cotangent) are positive
• Fourth quadrant (270 to 360 degrees): Only Cosine (and secant) are positive

The mnemonic CAST (reading anticlockwise from the fourth quadrant: Cosine, All, Sine, Tangent) tells you which functions are positive in each region. Every exam series tests this, so make it second nature.

The identities that hold everything together

Trigonometric identities are equations that are true for every angle, not just specific values. They're the backbone of nearly every proof and equation-solving question on the paper. You need to know three groups cold.

The Pythagorean family

The most fundamental identity is sin^2(x) + cos^2(x) = 1. If you divide every term by cos^2(x), you get tan^2(x) + 1 = sec^2(x). Divide instead by sin^2(x), and you get 1 + cot^2(x) = cosec^2(x). Three identities, one origin.

The quotient identities

tan(x) = sin(x) / cos(x) and cot(x) = cos(x) / sin(x). These two are your go-to substitutions whenever you need to rewrite everything in terms of sin and cos, which is the single most useful strategy for proving identities.

Proving trigonometric identities: a step-by-step approach

Identity proofs worry a lot of students, but they follow a reliable pattern once you know the tricks. The golden rule: work on one side only (usually the more complicated side) and transform it into the other.

1. Choose the messier side. If one side has sec, cosec, or cot, start there. Converting everything to sin and cos usually simplifies things.
2. Replace reciprocal functions. Write sec as 1/cos, cosec as 1/sin, cot as cos/sin, tan as sin/cos.
3. Look for Pythagorean substitutions. If you see 1 - sin^2, replace it with cos^2. If you see sec^2 - 1, that's tan^2.
4. Combine fractions. When you have two terms with different denominators, find a common denominator and add them.
5. Factor where possible. Difference of squares, common factors, and grouping all appear in trig proofs just as they do in algebra.

Solving trigonometric equations

Solving trig equations is where students lose the most marks, and it's almost never because they can't do the algebra. The marks vanish because they find one solution and forget the others. A question that says "solve for 0 <= x <= 360 degrees" usually has two, three, or even four answers.

The method that keeps you safe

1. Rearrange until you have a single trig function equals a value: sin(x) = 0.5, or tan(2x) = -1, or sec^2(x) = 4.
2. Find the principal value. Use your calculator or exact values to find the acute angle whose trig function gives that value. For sin(x) = 0.5, the principal value is 30 degrees.
3. Use CAST to find all angles in the given domain where the function has the correct sign. Sin is positive in quadrants 1 and 2, so x = 30 degrees and x = 180 - 30 = 150 degrees.
4. Adjust for compound angles. If the equation involves 2x or (x + 30), adjust the domain first. For sin(2x) = 0.5 in 0 to 360 degrees, you need 2x in 0 to 720 degrees. Find all values of 2x in that range, then halve them.

Trigonometric graphs: reading the wave

Graphs of trig functions are like fingerprints for oscillating behaviour. The general forms you need are y = a sin(bx) + c, y = a cos(bx) + c, and y = a tan(bx) + c. Each letter controls a different feature of the graph.

Think of it like tuning a radio. The amplitude is the volume (how tall the wave gets), the period is the frequency (how quickly it oscillates), and the vertical shift is the baseline (where the centre of the wave sits). Adjusting any one of these changes the shape of the graph without destroying the fundamental wave pattern.

Sketching transformed graphs

When a question asks you to sketch y = 3 sin(2x) + 1, break it apart:

• The amplitude is 3, so the wave goes 3 units above and below the centre line
• The period is 360/2 = 180 degrees, so one full cycle fits in 180 degrees instead of the usual 360
• The centre line is y = 1, so the wave oscillates between y = -2 and y = 4

Plot a few key points (start, first peak, centre crossing, trough, end of cycle) and connect them with a smooth curve. Examiners award marks for correct maximum and minimum values, correct period, and the correct shape. Labelling your axes with these values picks up those marks even if your curve isn't perfectly smooth.

The tangent graph

Tangent behaves differently from sine and cosine. It has no amplitude (it stretches to infinity), its period is 180 degrees (not 360), and it has vertical asymptotes wherever cosine equals zero. For y = tan(x), those asymptotes sit at x = 90, 270, 450 degrees, and so on. The graph passes through the origin and climbs steeply towards each asymptote.

For y = a tan(bx) + c, the value of a stretches the curve vertically, b compresses the period to 180/b degrees, and c shifts the whole graph up or down. The asymptotes shift too: they occur wherever bx equals 90, 270, and so on.

Common mistakes and how to dodge them

After years of marking IGCSE papers, certain errors show up again and again. Knowing what they are is half the battle.

Self-check questions

Try these without looking back at the notes. If you get stuck, the relevant section is noted in brackets.

1. Write sec(x) in terms of a single trig function. [Six functions table]
2. In which quadrants is tangent negative? [CAST diagram]
3. Prove that (1 - cos^2(x)) / sin(x) = sin(x). [Proving identities]
4. Solve cos(x) = -0.5 for 0 <= x <= 360 degrees. [Solving equations]
5. A graph has maximum value 5 and minimum value -1. What are the amplitude and vertical shift? [Graph parameters]
6. For y = 2 cos(3x) - 1, state the period and the range. [Graph transformations]
7. Starting from tan^2(x) + 1 = sec^2(x), derive the identity 1 + cot^2(x) = cosec^2(x). [Pythagorean family]

If questions 1, 2, and 5 feel automatic, your foundations are solid. If questions 3, 4, and 7 gave you trouble, spend extra time on identity proofs and equation solving before your exam. Those two skills together account for the bulk of the trigonometry marks on every IGCSE Additional Mathematics paper.