Cambridge IGCSE Additional Mathematics Notes 10: Trigonometry

Cambridge IGCSE Additional Mathematics 0606 Topic 10 covers sine, cosine, tangent, secant, cosecant and cotangent for angles of any magnitude. Candidates interpret transformed graphs in degrees or radians, use three supplied Pythagorean identities, solve equations over stated domains and prove relationships. Triangle rules and exact-value tables are not listed Topic 10 outcomes and are not treated as owners of this chapter.

1. Six trigonometric functions

Sine, cosine and tangent extend beyond acute triangles through their periodic graphs. The reciprocal functions are:

• sec x = 1/cos x;
• cosec x = 1/sin x;
• cot x = 1/tan x = cos x/sin x.

These definitions create restrictions. Sec is undefined where cos x = 0, cosec where sin x = 0, and cot where sin x = 0. A reciprocal function has magnitude at least 1 whenever it is defined for sec or cosec.

2. Signs and angles of any magnitude

Reference angles and quadrant signs generate solutions beyond the first quadrant. Sine is positive in quadrants I and II, cosine in I and IV, and tangent in I and III. Reciprocal functions share the sign of their underlying function.

For negative angles and angles beyond one revolution, use periodicity. Sine and cosine repeat every 360° or 2π; tangent and cotangent repeat every 180° or π.

3. Amplitude, period and vertical shift

For y = a sin bx + c or y = a cos bx + c, amplitude is |a| and the centre line is y = c. The period is 360°/|b| in degrees or 2π/|b| in radians.

For y = a tan bx + c, amplitude is not defined. Its period is 180°/|b| or π/|b|, and its centre line is y = c. Vertical asymptotes occur where bx reaches an odd multiple of 90° or π/2.

The official forms use positive integer a, simple fractional or integer b, and integer c. Fractions for b have denominators from the stated set 2, 3, 4, 6 or 8.

4. Sketching transformed graphs

Start with one period of the parent graph. Apply horizontal scaling from b, vertical scaling from a and vertical translation c. Mark centre line, maxima and minima for sine or cosine, zeros and clearly labelled tangent asymptotes.

Extend the pattern across the requested domain. Keep degree and radian labels consistent. A graph of tangent must not be joined across an asymptote.

Related graphs can be compared through shifts and sign changes, for example cosine as a shifted sine graph. State the domain and transformation rather than relying on a memorised picture alone.

5. Supplied identities

The formula list provides:

• sin²A + cos²A = 1;
• sec²A = 1 + tan²A;
• cosec²A = 1 + cot²A.

The second follows by dividing the first by cos²A; the third follows by dividing by sin²A. These derivations show their domain restrictions.

Combine them with reciprocal and quotient definitions to rewrite an equation in one function or prove a relationship.

6. Solving basic equations

Isolate the trigonometric function, find a reference angle and list every solution in the stated domain. A calculator inverse returns a principal value only. Use signs, symmetry and period to obtain the complete set.

For a transformed argument such as sin(x/3), solve first for the argument, apply its induced domain, then multiply back to recover x. Do not restrict the inner angle to the same interval as x without transforming the endpoints.

7. Equations involving reciprocal functions

Rewrite sec, cosec and cot in terms of sine, cosine or tangent when useful. For 4cot θ = tan θ, values where either side is undefined must be excluded. Multiplying by tan θ assumes it exists, so track those restrictions.

An equation quadratic in tan θ or sec θ can be solved algebraically, then each accepted function value generates angle solutions. Reject impossible values such as |sec θ| < 1 for real θ.

8. Homogeneous sine and cosine equations

An equation involving terms of the same degree in sin and cos can often be divided by cos² or sin² to form a quadratic in tan or cot. Before dividing, check separately whether the divisor can be zero and satisfy the original equation.

Alternatively use a Pythagorean identity to eliminate one squared function. Choose the route that preserves restrictions most transparently.

9. Proving identities

Begin with one side and transform it into the other. Do not manipulate both sides simultaneously, since that can assume the result. Convert reciprocal functions, factor, use a common denominator and apply a supplied identity where it simplifies the expression.

State restrictions if a step divides by sin x or cos x. A proof holds wherever both original sides are defined. Expanding everything immediately can hide the structure; look for conjugates and Pythagorean pairs.

10. Degrees, radians and accuracy

The domain may be in degrees or radians. Set calculator mode accordingly and express periods and solutions in the same unit. Exact multiples of π should remain exact when requested.

List endpoints only if they satisfy both the interval and the original equation. Order solutions and use the requested accuracy for non-exact values.

Worked example: solve a reciprocal equation on a domain

Solve 2sec²θ + tan θ - 3 = 0 for 0° ≤ θ < 360°. Use sec²θ = 1 + tan²θ to obtain 2(1 + tan²θ) + tan θ - 3 = 0, so 2tan²θ + tan θ - 1 = 0. Factorising gives (2tan θ - 1)(tan θ + 1) = 0. Thus tan θ = 1/2 or tan θ = -1. For tan θ = 1/2, θ ≈ 26.565° or 206.565°. For tan θ = -1, θ = 135° or 315°. All four values keep cosine non-zero, so sec is defined. To the nearest 0.1°, the solutions are 26.6°, 135.0°, 206.6° and 315.0°.

The final restriction check matters because rewriting sec in terms of tangent is valid only where the original reciprocal function exists.

Common misconceptions and how to correct them

• Treating sec x as cos⁻¹x. Sec is reciprocal cosine, not inverse cosine.
• Forgetting reciprocal-function restrictions. Exclude zeros of the denominator function.
• Assigning amplitude to tangent. Tangent has no finite maximum or minimum.
• Using 360°/b as tangent period. Tangent period is 180°/|b|.
• Leaving tangent asymptotes unlabelled. Cambridge explicitly requires x-coordinates.
• Joining tangent branches through an asymptote. They are disconnected.
• Giving only a principal inverse-calculator value. Generate all domain solutions.
• Using the x-domain directly for a transformed inner angle. Transform the interval.
• Dividing by sin or cos without checking zero. Test the excluded case separately.
• Accepting |sec θ| < 1. Real sec and cosec magnitudes are at least 1.
• Changing both sides during an identity proof. Transform one side into the other.
• Importing sine rule or cosine rule as Topic 10 ownership. They are not listed outcomes here.

Assessment guidance

State angle units and calculator mode. For transformed graphs, calculate period, centre line, extrema or tangent asymptotes and label the requested domain. When solving equations, show the identity or reciprocal rewrite, track undefined values and list all solutions in order. Transform the domain when the angle itself is scaled. For identity proofs, start from one side, justify reciprocal or Pythagorean substitutions and avoid cancelling expressions that may be zero without noting restrictions. Keep exact π forms where requested and round only final non-exact solutions. Do not add triangle-rule material that the official Topic 10 outcomes do not assign here.

Retrieval practice

Write the reciprocal and quotient definitions with their restrictions. Sketch transformed sine, cosine and tangent graphs in degrees and radians, labelling periods and asymptotes. Solve six equations involving all six functions over different domains, including a scaled argument and a quadratic in tangent. Prove three identities from one side only and state where each original expression is defined.

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Official source

Cambridge International, Additional Mathematics 0606 syllabus for examinations in 2025, 2026 and 2027.