Trigonometry: Identities and Equations

What you'll learn

This comprehensive guide covers all trigonometric identities and equations tested in the CXC CSEC Additional Mathematics examination. You will master the fundamental identities, learn to prove more complex relationships, and develop systematic techniques for solving trigonometric equations within specified domains. These skills are essential for Paper 2 Section I (pure mathematics) 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, such as sin²θ + cos²θ = 1.

Pythagorean identities — the three fundamental identities derived from the Pythagorean theorem: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Compound angle formulas — identities that express trigonometric functions of sums or differences of angles in terms of functions of individual angles, such as sin(A + B) = sinA cosB + cosA sinB.

Double angle formulas — special cases of compound angle formulas where both angles are equal, such as sin2A = 2sinA cosA and cos2A = cos²A - sin²A.

Principal value — the value of an angle in the restricted domain that satisfies a trigonometric equation, typically 0° ≤ θ ≤ 360° or 0 ≤ θ ≤ 2π radians.

Reciprocal identities — relationships between the basic trigonometric ratios and their reciprocals: secθ = 1/cosθ, cosecθ = 1/sinθ, and cotθ = 1/tanθ.

Factor formula — identities that convert sums or differences of sines and cosines into products, useful for solving equations.

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

Core concepts

The fundamental trigonometric identities

The Pythagorean identity sin²θ + cos²θ = 1 forms the foundation of trigonometric manipulation. From this, you can derive two additional forms by dividing through by cos²θ or sin²θ:

Dividing by cos²θ:

• tan²θ + 1 = sec²θ

Dividing by sin²θ:

• 1 + cot²θ = cosec²θ

The quotient identities connect different trigonometric ratios:

• tanθ = sinθ/cosθ
• cotθ = cosθ/sinθ

The reciprocal identities are:

• secθ = 1/cosθ
• cosecθ = 1/sinθ
• cotθ = 1/tanθ

Remember that tanθ·cotθ = 1 for all values where both are defined.

Proving trigonometric identities

When proving identities at CSEC level, work systematically from one side of the equation to reach the other side. Never work on both sides simultaneously during your proof.

Standard approach:

1. Start with the more complex side
2. Express all functions in terms of sine and cosine when stuck
3. Look for opportunities to use Pythagorean identities
4. Combine fractions over a common denominator
5. Factor or expand expressions strategically

For example, to prove (1 - cos²θ)/sinθ = sinθ:

Left side: (1 - cos²θ)/sinθ = sin²θ/sinθ = sinθ = Right side ✓

The key substitution used sin²θ for (1 - cos²θ) based on the Pythagorean identity.

Compound angle formulas

These formulas must be memorized for the CSEC examination:

Addition formulas:

• sin(A + B) = sinA cosB + cosA sinB
• cos(A + B) = cosA cosB - sinA sinB
• tan(A + B) = (tanA + tanB)/(1 - tanA tanB)

Subtraction formulas:

• sin(A - B) = sinA cosB - cosA sinB
• cos(A - B) = cosA cosB + sinA sinB
• tan(A - B) = (tanA - tanB)/(1 + tanA tanB)

These formulas are frequently tested in questions asking you to find exact values. For instance, sin15° can be calculated as sin(45° - 30°) using known values of sine and cosine at 30° and 45°.

Double angle formulas

Setting B = A in the compound angle formulas gives the double angle formulas:

For sine:

• sin2A = 2sinA cosA

For cosine (three equivalent forms):

• cos2A = cos²A - sin²A
• cos2A = 2cos²A - 1
• cos2A = 1 - 2sin²A

For tangent:

• tan2A = 2tanA/(1 - tan²A)

The three forms of cos2A are particularly useful. Choose the form that matches the information given in the question. The second and third forms can be rearranged to express cos²A or sin²A in terms of cos2A, which is useful for integration at advanced level.

Solving trigonometric equations

CSEC questions typically specify the domain, such as 0° ≤ θ ≤ 360° or -180° < θ ≤ 180°. Follow this systematic approach:

Method for solving trigonometric equations:

1. Rearrange the equation to isolate one trigonometric function
2. Find the principal value using a calculator or exact values
3. Apply symmetry to find other solutions in the specified domain
4. Check all solutions satisfy the original equation

For equations like sin²θ - 5sinθ + 6 = 0, treat this as a quadratic in sinθ:

• Let x = sinθ
• Solve x² - 5x + 6 = 0
• Factor: (x - 2)(x - 3) = 0
• Since sinθ cannot exceed 1, reject sinθ = 2 and sinθ = 3
• No solutions exist

For equations involving multiple angles like sin2θ = cosθ:

• Use the double angle formula: 2sinθ cosθ = cosθ
• Rearrange: 2sinθ cosθ - cosθ = 0
• Factor: cosθ(2sinθ - 1) = 0
• Solve cosθ = 0 or sinθ = 0.5 separately

Remember the CAST diagram (or "All Stations To Central" mnemonic used in Caribbean schools) to determine which quadrants give positive values for each trigonometric ratio.

The auxiliary angle method

Expressions of the form a sinθ + b cosθ can be rewritten as R sin(θ + α) or R cos(θ - α), where:

• R = √(a² + b²)
• tanα = b/a (for sine form) or tanα = a/b (for cosine form)

This technique is essential for finding maximum and minimum values, since the range of R sin(θ + α) is [-R, R].

Example application: A telecommunications tower in Port of Spain receives signal strength S = 3sinθ + 4cosθ watts. To find maximum signal strength:

• R = √(3² + 4²) = √25 = 5
• Maximum signal strength = 5 watts

At CSEC level, you may be asked to express in the required form and then solve equations or find extreme values.

Worked examples

Example 1: Proving an identity (4 marks)

Question: Prove that (sinθ + cosθ)² = 1 + 2sinθ cosθ

Solution:

Start with the left side (more complex):

LHS = (sinθ + cosθ)²

Expand the bracket: = sin²θ + 2sinθ cosθ + cos²θ

Rearrange terms: = sin²θ + cos²θ + 2sinθ cosθ

Apply Pythagorean identity (sin²θ + cos²θ = 1): = 1 + 2sinθ cosθ

= RHS ✓

Hence proved.

Mark allocation: Correct expansion (1), recognition of Pythagorean identity (1), correct substitution (1), clear conclusion (1)

Example 2: Solving a trigonometric equation (6 marks)

Question: Solve the equation 2cos²x - sinx = 1 for 0° ≤ x ≤ 360°.

Solution:

Use the identity sin²x + cos²x = 1, so cos²x = 1 - sin²x

Substitute into the equation: 2(1 - sin²x) - sinx = 1

Expand: 2 - 2sin²x - sinx = 1

Rearrange to standard quadratic form: 2sin²x + sinx - 1 = 0

Factor: (2sinx - 1)(sinx + 1) = 0

Therefore: sinx = 1/2 or sinx = -1

For sinx = 1/2: x = 30° (first quadrant) x = 180° - 30° = 150° (second quadrant, since sine is positive)

For sinx = -1: x = 270°

Solutions: x = 30°, 150°, 270°

Mark allocation: Correct substitution using Pythagorean identity (1), rearranging to quadratic form (1), correct factorization (1), finding sinx = 1/2 solutions (2), finding sinx = -1 solution (1)

Example 3: Application of compound angle formula (5 marks)

Question: A cargo ship leaving Kingston harbour follows a course where its distance D km from port after t hours is given by D = 15sin(t + 60°).

Given that sin60° = √3/2 and cos60° = 1/2, express D in the form a sint + b cost, where a and b are constants to be determined.

Solution:

Use the compound angle formula: sin(A + B) = sinA cosB + cosA sinB

D = 15sin(t + 60°)

Apply the formula with A = t and B = 60°: D = 15(sint cos60° + cost sin60°)

Substitute the given values: D = 15(sint × 1/2 + cost × √3/2)

Expand: D = 15sint/2 + 15√3cost/2

Simplify: D = 7.5sint + 7.5√3cost

Therefore: a = 7.5, b = 7.5√3 (or 15/2 and 15√3/2 in exact form)

Mark allocation: Correct identification of compound angle formula (1), correct application (1), substitution of given values (1), correct simplification (1), final answer in required form (1)

Common mistakes and how to avoid them

• Confusing identities with equations: An identity is true for all values; an equation is only true for specific values. When solving equations, always specify the domain and find all solutions within it.

• Incorrect algebraic manipulation: When squaring both sides of a trigonometric equation, you may introduce extraneous solutions. Always substitute your answers back into the original equation to verify.

• Missing solutions: After finding the principal value, students often forget to use symmetry properties to find all solutions in the given domain. For 0° ≤ θ ≤ 360°, sine has two solutions (except at 90° and 270°), cosine has two solutions (except at 0°, 180°, and 360°), and tangent has two solutions.

• Sign errors with compound angles: Pay careful attention to the signs in formulas. Notice that cos(A + B) has a minus sign while cos(A - B) has a plus sign—opposite to what intuition might suggest.

• Working on both sides simultaneously when proving: In identity proofs, you must transform one side into the other using valid algebraic steps. Working on both sides is not accepted in CSEC marking schemes.

• Calculator mode errors: Ensure your calculator is in degree mode when angles are given in degrees, and radian mode for radians. CSEC questions will specify which to use.

Exam technique for "Trigonometry: Identities and Equations"

• Command word "Prove": Show every algebraic step clearly. Start from one side (usually the more complex) and work systematically to reach the other side. Write "LHS" and "RHS" or use the ∴ symbol for "therefore" to structure your proof. Each line should follow logically from the previous one. Worth typically 3-5 marks.

• Command word "Solve": State the domain you're working in, find all solutions, and present them in ascending order. Show the principal value calculation, then use trigonometric properties to find remaining solutions. Verify solutions when time permits. Worth typically 4-7 marks depending on complexity.

• Express/Write in the form: This command requires you to manipulate the expression into a specific format, usually involving auxiliary angles or double angle formulas. Show the method used to find constants, not just the final answer. Worth 3-5 marks.

• Time management: Identity proofs and equation solving typically appear as part (a) and part (b) of the same question. If stuck on a proof, move to the equation-solving part which often uses the identity you were asked to prove. Return to the proof if time allows.

Quick revision summary

Master the three Pythagorean identities and reciprocal relationships. Memorize compound angle formulas for sin(A ± B), cos(A ± B), and tan(A ± B), then derive double angle formulas as special cases. When proving identities, work from one side only, expressing everything in sine and cosine when stuck. For equations, rearrange to isolate one function, find the principal value, then use CAST to find all solutions in the domain. Factor expressions involving sin²θ or cos²θ as quadratics. The auxiliary angle method converts a sinθ + b cosθ to R sin(θ + α) for finding maximum values.