Trigonometry: Identities and Equations

What you'll learn

Trigonometry: Identities and Equations forms a substantial component of the CIE IGCSE Additional Mathematics syllabus, building on the basic trigonometric ratios studied at ordinary level. This topic requires you to prove trigonometric identities, solve trigonometric equations within specified intervals, and apply both skills to complex problems involving compound and double angle formulae. Examiners consistently test your ability to manipulate identities algebraically and solve equations systematically, often combining these skills in multi-step questions worth 8-12 marks.

Key terms and definitions

Trigonometric identity — an equation involving trigonometric functions that is true for all values of the variable for which both sides are defined.

Pythagorean identity — the fundamental relationship sin²θ + cos²θ = 1, from which related identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are derived.

Compound angle formulae — expressions for sin(A ± B), cos(A ± B), and tan(A ± B) in terms of trigonometric functions of A and B separately.

Double angle formulae — special cases of compound angle formulae where both angles are equal, giving expressions for sin 2A, cos 2A, and tan 2A.

Principal value — the primary solution to a trigonometric equation, typically the smallest non-negative angle that satisfies the equation.

General solution — the complete set of all angles satisfying a trigonometric equation, expressed using the periodicity of trigonometric functions.

Auxiliary angle method — a technique for expressing a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α), where R > 0.

Core concepts

The fundamental trigonometric identities

The Pythagorean identity sin²θ + cos²θ = 1 serves as the foundation for proving most other identities. From this, two additional identities emerge by dividing throughout by cos²θ or sin²θ:

• Dividing by cos²θ: tan²θ + 1 = sec²θ
• Dividing by sin²θ: 1 + cot²θ = cosec²θ

The reciprocal identities connect the basic ratios with their reciprocals:

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ = cos θ/sin θ

The quotient identity tan θ = sin θ/cos θ appears frequently in CIE IGCSE Additional Mathematics papers, particularly when simplifying complex expressions.

Proving trigonometric identities

Identity proofs require rigorous algebraic manipulation. The standard approach follows these steps:

1. Start with the more complex side — typically the left-hand side if both sides appear equally complex
2. Work towards the simpler side — manipulate only one side of the equation
3. Use substitution — replace functions using known identities (commonly tan θ = sin θ/cos θ)
4. Simplify systematically — combine fractions, factorise, or expand as needed
5. Arrive at the target expression — demonstrate both sides are identical

Critical techniques for CIE papers:

• Express everything in terms of sin θ and cos θ when the identity involves multiple different functions
• Multiply by conjugates when dealing with expressions containing (1 ± sin θ) or (1 ± cos θ)
• Factor common terms before attempting to simplify complex fractions
• Work with one side only — never perform operations on both sides simultaneously, as this is not a valid proof technique

Compound angle formulae

The addition formulae must be memorised exactly as they appear frequently:

• sin(A + B) = sin A cos B + cos A sin B
• sin(A − B) = sin A cos B − cos A sin B
• cos(A + B) = cos A cos B − sin A sin B
• cos(A − B) = cos A cos B + sin A sin B
• tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
• tan(A − B) = (tan A − tan B)/(1 + tan A tan B)

These formulae enable calculation of exact values for angles like 15°, 75°, and 105° by expressing them as sums or differences of 30°, 45°, and 60°.

Double angle formulae

Setting B = A in the compound angle formulae yields the double angle formulae:

For sine:

• sin 2A = 2 sin A cos A

For cosine (three equivalent forms):

• cos 2A = cos²A − sin²A
• cos 2A = 2cos²A − 1
• cos 2A = 1 − 2sin²A

For tangent:

• tan 2A = 2 tan A/(1 − tan²A)

The three forms of cos 2A prove particularly useful in different contexts. Examiners often require you to select the appropriate form based on the information provided in the question.

Solving trigonometric equations

CIE IGCSE Additional Mathematics papers require solutions within a specified interval, typically 0° ≤ θ ≤ 360° or 0 ≤ x ≤ 2π radians.

Standard equation solving procedure:

1. Rearrange to standard form — isolate the trigonometric function
2. Find the principal value — use inverse functions on your calculator
3. Identify the quadrants — determine which quadrants contain solutions based on the sign
4. Apply CAST diagram — All positive (1st), Sin positive (2nd), Tan positive (3rd), Cos positive (4th)
5. Calculate all solutions — use symmetry properties of trigonometric graphs
6. Verify the interval — ensure all solutions fall within the specified range

For sin θ = k:

• If k > 0: solutions in 1st and 2nd quadrants (θ and 180° − θ)
• If k < 0: solutions in 3rd and 4th quadrants (180° + θ and 360° − θ)

For cos θ = k:

• If k > 0: solutions in 1st and 4th quadrants (θ and 360° − θ)
• If k < 0: solutions in 2nd and 3rd quadrants (180° − θ and 180° + θ)

For tan θ = k:

• Solutions in 1st and 3rd quadrants if k > 0, or 2nd and 4th if k < 0
• General pattern: θ and θ + 180°

Equations requiring algebraic manipulation

More complex equations require factorisation or substitution before solving:

Type 1: Quadratic in one function Example: 2sin²θ + 3sin θ − 2 = 0

• Factorise: (2sin θ − 1)(sin θ + 2) = 0
• Solve each factor separately, rejecting impossible solutions (sin θ = −2)

Type 2: Multiple angle equations Example: cos 2θ = sin θ for 0° ≤ θ ≤ 360°

• Substitute using double angle formula: 1 − 2sin²θ = sin θ
• Rearrange to quadratic: 2sin²θ + sin θ − 1 = 0
• Solve the quadratic equation

Type 3: Mixed function equations

• Convert all terms to the same function using identities
• Commonly convert tan θ to sin θ/cos θ or use sin²θ + cos²θ = 1

The auxiliary angle method

Expressions of the form a cos θ + b sin θ can be rewritten as a single trigonometric function, simplifying equation solving and finding maximum/minimum values.

Standard form: a cos θ + b sin θ = R cos(θ − α) or R sin(θ + α)

To find R and α:

• R = √(a² + b²)
• tan α = b/a (when using R cos(θ − α))
• tan α = a/b (when using R sin(θ + α))

This technique appears regularly in CIE papers, particularly in questions requiring maximum or minimum values, since the range of R cos(θ − α) is [−R, R].

Worked examples

Example 1: Proving an identity

Question: Prove the identity (sin θ + cos θ)² = 1 + sin 2θ

Solution:

Expand the left-hand side: LHS = (sin θ + cos θ)² = sin²θ + 2sin θ cos θ + cos²θ

Rearrange using the Pythagorean identity: = (sin²θ + cos²θ) + 2sin θ cos θ = 1 + 2sin θ cos θ

Apply the double angle formula for sine: = 1 + sin 2θ = RHS ✓

The identity is proved.

Example 2: Solving a trigonometric equation

Question: Solve the equation 4cos²x − 4sin x − 1 = 0 for 0° ≤ x ≤ 360°

Solution:

Substitute cos²x = 1 − sin²x: 4(1 − sin²x) − 4sin x − 1 = 0 4 − 4sin²x − 4sin x − 1 = 0 −4sin²x − 4sin x + 3 = 0 4sin²x + 4sin x − 3 = 0

Factorise: (2sin x + 3)(2sin x − 1) = 0

From 2sin x + 3 = 0: sin x = −3/2 (impossible, as −1 ≤ sin x ≤ 1)

From 2sin x − 1 = 0: sin x = 1/2

Principal value: x = 30°

Since sin x is positive, solutions are in the 1st and 2nd quadrants: x = 30° or x = 180° − 30° = 150°

Answer: x = 30°, 150°

Example 3: Using the auxiliary angle method

Question: Express 3cos θ + 4sin θ in the form R cos(θ − α), where R > 0 and 0° < α < 90°. Hence find the maximum value of 3cos θ + 4sin θ.

Solution:

R = √(3² + 4²) = √(9 + 16) = √25 = 5

tan α = 4/3 α = tan⁻¹(4/3) = 53.1° (to 1 d.p.)

Therefore: 3cos θ + 4sin θ = 5cos(θ − 53.1°)

The maximum value of cos(θ − 53.1°) is 1, occurring when θ − 53.1° = 0°, i.e., θ = 53.1°.

Maximum value = 5 × 1 = 5

Common mistakes and how to avoid them

• Mistake: Treating identities like equations and performing operations on both sides during proofs. Correction: Work with one side only, transforming it step-by-step until it matches the other side exactly.

• Mistake: Forgetting the second solution when solving basic trigonometric equations, particularly missing the obtuse angle for sin θ = k where k > 0. Correction: Always use the CAST diagram to identify which quadrants contain solutions, and apply the appropriate symmetry formula.

• Mistake: Dividing both sides of an equation by a trigonometric function (e.g., dividing by cos θ), potentially losing solutions where that function equals zero. Correction: Rearrange to factorise instead; never divide by a variable expression.

• Mistake: Using the wrong form of cos 2A for the problem context. Correction: If the question involves only sin A, use cos 2A = 1 − 2sin²A; if only cos A, use cos 2A = 2cos²A − 1; if mixed, use cos 2A = cos²A − sin²A.

• Mistake: Confusing compound angle formulae, particularly the signs in cos(A + B) = cos A cos B − sin A sin B versus sin(A + B) = sin A cos B + cos A sin B. Correction: Note that cosine has the "opposite" sign pattern; memorise formulae through regular practice.

• Mistake: In auxiliary angle questions, calculating R or α incorrectly or using the wrong form (cos vs sin). Correction: Double-check that R = √(a² + b²) and verify which form the question specifies; sketch the right-angled triangle to confirm tan α.

Exam technique for Trigonometry: Identities and Equations

• Identity proofs (4-6 marks): Show every step of your working clearly. Write "LHS =" and "RHS =" to indicate which side you're manipulating. State which identity you're applying at each stage (e.g., "Using sin²θ + cos²θ = 1"). The final line must show the two sides are identical.

• Equation solving (5-8 marks): Always state the principal value first, then systematically find all solutions in the given interval. Present answers in ascending order. For radian answers, give exact values in terms of π where possible, or 3 significant figures if decimal.

• "Show that" questions: These require you to arrive at the given answer through clear working. Examiners award marks for method even if you know the answer. Work must be detailed enough to demonstrate every logical step.

• Mixed questions combining identities and equations: Read the entire question first to identify the overall strategy. Part (a) often establishes an identity needed for part (b)'s equation. Use earlier results explicitly by referring to them: "From part (a), we have..."

Quick revision summary

Master the three Pythagorean identities and all compound angle formulae. For identity proofs, manipulate one side only using systematic substitution and algebraic techniques. When solving equations, always find the principal value first, then use CAST to identify all solutions in the specified interval. Remember the three forms of cos 2A and select appropriately. The auxiliary angle method converts a cos θ + b sin θ to R cos(θ − α) where R = √(a² + b²) and tan α = b/a. Practice factorising quadratic trigonometric expressions and converting mixed-function equations into single functions before solving.