IB Maths AA Analysis and Approaches How to Solve: IB AA HL/SL – Trigonometry (Compound/Double Angle, Solving Equations, Graph Transformations)

How to Solve: IB AA HL/SL – Trigonometry (Compound/Double Angle, Solving Equations, Graph Transformations)

Complete Guide

Introduction

"Mastering compound and double-angle trigonometry doesn’t just get you 6–8 marks on your IB Math AA exam—it’s the key to modeling wave interference in Physics, optimizing chemical reaction rates, and even predicting economic cycles. One question on this topic can be the difference between a 5 and a 7."

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you can: 1. Recall exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, 90° (and radians). 2. Solve basic trig equations (e.g., sin x = 0.5) and find all solutions in a given interval. 3. Graph basic trig functions (sin x, cos x, tan x) and identify amplitude, period, and phase shifts.

KEY TERMS & FORMULAS

1. Compound Angle Formulas

(Given on exam sheet, but memorize for speed!) - sin(A ± B) = sin A cos B ± cos A sin B - A, B: Angles in degrees or radians. - cos(A ± B) = cos A cos B ∓ sin A sin B - Note: The sign flips for cosine! - tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B) - Undefined when denominator = 0 (e.g., tan 90°).

2. Double Angle Formulas

(MEMORISE THIS!) - sin(2A) = 2 sin A cos A - cos(2A) = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A - Three versions—use the one that matches the given info. - tan(2A) = 2 tan A / (1 – tan²A)

3. Graph Transformations

(MEMORISE THIS!) For y = a sin(b(x – c)) + d or y = a cos(b(x – c)) + d: - Amplitude (a): Vertical stretch/compression. |a| = max height from midline. - Period (2π/|b|): Horizontal stretch/compression. b = 2π/period. - Phase shift (c): Horizontal shift. +c = shift right, –c = shift left. - Vertical shift (d): Moves midline up/down. d = new midline.

STEP-BY-STEP METHOD

Part 1: Solving Trig Equations with Compound/Double Angles

Step 1: Identify the angle structure. - Is it sin(2x), cos(x + 30°), tan(3x – π/4)? Circle the angle.

Step 2: Apply the correct formula. - Double angle? Use sin(2A), cos(2A), or tan(2A). - Compound angle? Use sin(A ± B), cos(A ± B), or tan(A ± B).

Step 3: Simplify the equation. - Expand using the formula, then combine like terms. - If possible, rewrite in terms of a single trig function (e.g., 2 sin x cos x = sin(2x)).

Step 4: Solve for the angle. - Isolate the trig function (e.g., sin(2x) = 0.5). - Find the principal solution (first solution in the unit circle). - Find all solutions in the given interval using +2πn (radians) or +360°n (degrees).

Step 5: Check for extraneous solutions. - Plug solutions back into the original equation to verify.

Part 2: Graph Transformations

Step 1: Identify the base function. - Is it sin x, cos x, or tan x?

Step 2: List the transformations in order. 1. Vertical stretch/compression (a) → Amplitude = |a|. 2. Horizontal stretch/compression (b) → Period = 2π/|b|. 3. Phase shift (c) → Shift right by c if (x – c), left by c if (x + c). 4. Vertical shift (d) → New midline = d.

Step 3: Sketch the graph. - Start with the base function. - Apply transformations one at a time in the order above. - Label key points (max, min, intercepts, asymptotes for tan).

WORKED EXAMPLES

Example 1 – Basic: Solve sin(2x) = √3/2 for 0 ≤ x ≤ 2π

Step 1: Identify the angle: 2x. Step 2: Use double-angle formula (already simplified). Step 3: Solve sin(2x) = √3/2. - Principal solutions: 2x = π/3 and 2x = 2π/3. - General solutions: 2x = π/3 + 2πn or 2x = 2π/3 + 2πn. Step 4: Divide by 2: x = π/6 + πn or x = π/3 + πn. Step 5: Find solutions in 0 ≤ x ≤ 2π: - x = π/6, 7π/6, π/3, 4π/3.

What we did and why: We used the double-angle identity (though it was already in double-angle form) and solved for x by finding all possible angles in the interval. The +2πn ensures we don’t miss any solutions.

Example 2 – Medium: Solve cos(x + π/4) = –1/2 for –π ≤ x ≤ π

Step 1: Identify the angle: x + π/4. Step 2: Use compound angle formula (already simplified). Step 3: Solve cos(x + π/4) = –1/2. - Principal solutions: x + π/4 = 2π/3 or x + π/4 = 4π/3. - General solutions: x + π/4 = 2π/3 + 2πn or x + π/4 = 4π/3 + 2πn. Step 4: Subtract π/4: - x = 2π/3 – π/4 + 2πn = 5π/12 + 2πn - x = 4π/3 – π/4 + 2πn = 13π/12 + 2πn Step 5: Find solutions in –π ≤ x ≤ π: - x = 5π/12 (from n = 0) - x = 13π/12 – 2π = –11π/12 (from n = –1)

What we did and why: We treated x + π/4 as a single angle, solved for it, then isolated x. The +2πn ensures we check all possible angles, and we adjusted n to fit the interval.

Example 3 – Exam-Style: Sketch y = –2 cos(3x – π/2) + 1

Step 1: Base function: cos x. Step 2: Transformations: 1. Vertical stretch by 2 → Amplitude = 2. 2. Reflection over x-axis → Negative sign (–2). 3. Horizontal compression by 3 → Period = 2π/3. 4. Phase shift right by π/6 → 3x – π/2 = 3(x – π/6). 5. Vertical shift up by 1 → Midline = y = 1. Step 3: Sketch: - Start with cos x (period 2π, amplitude 1, midline y = 0). - Apply vertical stretch/reflection: –2 cos x (amplitude 2, upside-down). - Apply horizontal compression: –2 cos(3x) (period 2π/3). - Apply phase shift: –2 cos(3(x – π/6)) (shift right by π/6). - Apply vertical shift: –2 cos(3x – π/2) + 1 (midline y = 1). Key points: - Max: y = 3 (when cos = –1) - Min: y = –1 (when cos = 1) - Intercepts: Solve –2 cos(3x – π/2) + 1 = 0 → cos(3x – π/2) = 0.5.

What we did and why: We broke the transformation into steps to avoid mistakes. The phase shift is tricky—always factor out the coefficient of x first!

COMMON MISTAKES

1. MISTAKE: Forgetting the ±2πn (or ±360°n) when solving equations.
   WHY IT HAPPENS: Students stop at the principal solution.
   CORRECT APPROACH: Always write the general solution first, then find specific solutions in the interval.

2. MISTAKE: Mixing up the signs in cos(A ± B).
   WHY IT HAPPENS: The formula has ∓, which is easy to misread.
   CORRECT APPROACH: Remember: "cosine is cozy—it likes to stay together" (cos cos and sin sin have the same sign).

3. MISTAKE: Incorrectly applying phase shifts (e.g., y = sin(x – π/2) shifts right, not left).
   WHY IT HAPPENS: Confusing –c with left shift.
   CORRECT APPROACH: Think: "Inside the function, opposite sign. Outside, same sign."

4. MISTAKE: Using the wrong version of cos(2A) (e.g., using 2cos²A – 1 when sin A is given).
   WHY IT HAPPENS: Not matching the formula to the given info.
   CORRECT APPROACH: If sin A is given, use 1 – 2sin²A. If cos A is given, use 2cos²A – 1.

5. MISTAKE: Forgetting to check for extraneous solutions (e.g., tan x undefined at π/2).
   WHY IT HAPPENS: Not verifying solutions in the original equation.
   CORRECT APPROACH: Always plug solutions back in!

EXAM TRAPS

1. TRAP: Giving an equation like sin(2x) = 1 and asking for solutions in 0 ≤ x ≤ 4π.
   HOW TO SPOT IT: The interval is larger than 2π, so you’ll need more solutions.
   HOW TO AVOID IT: Find all solutions in 0 ≤ 2x ≤ 8π, then divide by 2.

2. TRAP: Asking for cos(2x) in terms of sin x or cos x but not specifying which version to use.
   HOW TO SPOT IT: The question says "express in terms of...".
   HOW TO AVOID IT: Use the version that matches the given info (e.g., if sin x is given, use 1 – 2sin²x).

3. TRAP: Graphing y = 3 sin(πx + π/2) and asking for the period.
   HOW TO SPOT IT: The coefficient of x is π, not 1.
   HOW TO AVOID IT: Period = 2π/|b| → 2π/π = 2.

1-MINUTE RECAP

"Listen up—this is your 60-second survival guide for trigonometry on exam day. First, memorize the double-angle formulas: sin(2A) = 2 sin A cos A, and cos(2A) has three versions—pick the one that fits the question. For compound angles, remember the sign flips for cosine. When solving equations, always find the general solution first, then narrow it down to the interval. For graphs, start with the base function, then apply transformations in order: amplitude, period, phase shift, vertical shift. And watch out for traps—examiners love giving intervals larger than 2π or hiding phase shifts in the equation. You’ve got this—now go ace that exam!