The Six Trigonometric Functions Explained: IGCSE Mathematics - Additional Complete Breakdown

What the six trigonometric functions are and why they matter

The IGCSE Additional Mathematics syllabus (0606) requires fluency with six trigonometric functions, not just the three from standard Mathematics. Sine, cosine and tangent form the primary trio. Cosecant, secant and cotangent are their reciprocals. Together, these six functions underpin trigonometric equations, identities and calculus problems across both Paper 1 and Paper 2.

This breakdown covers definitions, exact values, graphs, identities and equation-solving techniques. Each section builds on the last, so working through in order is recommended.

Key facts

• Six functions total: sin, cos, tan (primary) and cosec, sec, cot (reciprocal)
• Reciprocal relationships: cosec x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x = cos x/sin x
• Three Pythagorean identities: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x
• Exact values required: for 0, 30, 45, 60 and 90 degrees (and their radian equivalents)
• CAST diagram: determines which functions are positive in each quadrant

Definitions from the unit circle

Place a point P on the unit circle (radius 1, centre at the origin). If the angle measured anticlockwise from the positive x-axis is x, then the coordinates of P are (cos x, sin x). The tangent function is the ratio: tan x = sin x / cos x.

The three reciprocal functions follow directly.

The "undefined" column matters in exams. When a question asks for the domain or range of a reciprocal function, these exclusions form the answer.

Exact values for standard angles

Cambridge examiners expect these values from memory. No marks are awarded for decimal approximations when exact form is specified.

Graphs: shape, period and range

Recognising the graph of each function is a frequent exam requirement. The table below summarises the essential properties.

The reciprocal functions share the period of their parent but never cross the region between -1 and 1. Their vertical asymptotes occur exactly where the parent function equals zero.

The Pythagorean identities

The fundamental identity sin²x + cos²x = 1 generates two further identities by division.

1. Divide by cos²x: tan²x + 1 = sec²x
2. Divide by sin²x: 1 + cot²x = cosec²x

These identities serve two purposes in IGCSE Additional Mathematics exams. First, they simplify expressions: replacing sec²x with 1 + tan²x can reduce an equation to a quadratic in tan x. Second, they prove other identities: most "show that" questions require substituting one identity into another and simplifying algebraically.

Worked example 1: Simplifying with an identity

Simplify (sec²x - 1) / tan x.

1. Replace sec²x - 1 using the identity: sec²x - 1 = tan²x
2. The expression becomes tan²x / tan x = tan x

The answer is tan x. Examiners award method marks for citing the identity explicitly before simplifying.

Worked example 2: Proving an identity

Prove that cosec²x - cot²x = 1.

1. Start from the left side. Use the identity 1 + cot²x = cosec²x, rearranged: cosec²x - cot²x = 1
2. The left side equals the right side directly by the third Pythagorean identity.

For a longer proof question, work on one side only and transform it into the other. Never manipulate both sides simultaneously.

The CAST diagram and solving equations

Trigonometric equations in the IGCSE Additional Mathematics course almost always require all solutions within a specified interval, typically 0 to 360 degrees. The CAST diagram determines which quadrants yield positive values.

• Quadrant I (0 to 90): All functions positive
• Quadrant II (90 to 180): Only Sine (and cosec) positive
• Quadrant III (180 to 270): Only Tangent (and cot) positive
• Quadrant IV (270 to 360): Only Cosine (and sec) positive

The mnemonic reads "All Students Take Calculus" (or any regional variant that sticks). The reciprocal functions follow the same sign pattern as their parent: cosec is positive wherever sin is positive, sec wherever cos is positive, cot wherever tan is positive.

Worked example 3: Solving a basic equation

Solve sin x = 0.5 for 0 ≤ x ≤ 360.

1. The reference angle is sin^-1(0.5) = 30 degrees
2. Sin is positive in Quadrants I and II
3. Quadrant I: x = 30 degrees
4. Quadrant II: x = 180 - 30 = 150 degrees

Solutions: x = 30 and x = 150.

Worked example 4: Equation leading to a quadratic

Solve 2cos²x - 3cos x + 1 = 0 for 0 ≤ x ≤ 360.

1. Let u = cos x. The equation becomes 2u² - 3u + 1 = 0
2. Factorise: (2u - 1)(u - 1) = 0
3. So u = 1/2 or u = 1
4. cos x = 1/2 gives x = 60 (Q I) and x = 300 (Q IV)
5. cos x = 1 gives x = 0 and x = 360

Solutions: x = 0, 60, 300 and 360. Candidates who stop after the quadratic step and forget to convert back to angles lose the final marks.

Worked example 5: Using an identity to solve

Solve sec²x = 3 + tan x for 0 ≤ x ≤ 360.

1. Replace sec²x with 1 + tan²x: 1 + tan²x = 3 + tan x
2. Rearrange: tan²x - tan x - 2 = 0
3. Factorise: (tan x - 2)(tan x + 1) = 0
4. tan x = 2: reference angle = tan^-1(2) = 63.4 degrees. Positive in Q I and Q III: x = 63.4 and x = 243.4
5. tan x = -1: reference angle = 45 degrees. Negative in Q II and Q IV: x = 135 and x = 315

Solutions: x = 63.4, 135, 243.4 and 315 (to 1 decimal place). This pattern - identity substitution, then quadratic, then CAST - accounts for a large share of the trigonometric equation marks on both papers.

Common exam question patterns

Three question types dominate the IGCSE Additional Mathematics papers for this topic.

A fourth type combines trigonometry with calculus: differentiate y = sec x or find the stationary points of y = 2sin x + cos 2x. These rely on the derivative rules covered in the calculus section, but the equation-solving step uses the methods above.

Common errors

Self-check questions

1. Write down the exact value of sec 60 and cosec 30.
2. Simplify (1 - cos²x) / sin x.
3. Solve 2sin²x - sin x - 1 = 0 for 0 ≤ x ≤ 360.
4. Prove that (1 + tan²x) cos²x = 1.
5. Solve cot²x + cosec x = 1 for 0 ≤ x ≤ 360.

Exam strategy for trigonometric function questions

Start every trig equation by identifying which identity, if any, reduces the number of different functions to one. A mixed equation in sec and tan calls for sec²x = 1 + tan²x. A mixed equation in cosec and cot calls for cosec²x = 1 + cot²x. Once the equation contains only one trigonometric function, it either solves directly or factors as a quadratic.

For proof questions, convert everything to sin and cos as a fallback strategy. It isn't always the shortest route, but it is reliable and earns full marks. Write each step on a separate line. Examiners follow the algebraic chain link by link; compressed working obscures the method and risks losing marks even when the final line is correct.

When finding multiple solutions, sketch the relevant graph across the given domain as a quick check. The number of intersections between the curve and the horizontal line y = k should match the number of solutions you found. If it doesn't, a solution has been missed or invented.