Compound Angle Formulas in Trigonometry

Overview

This lesson introduces compound angle formulas in grade 12 trigonometry, focusing on identifying, expanding, and simplifying expressions using these formulas and special angles.

Compound Angle Formulas

• Compound angle formulas involve sine and cosine of sum or difference of two angles.
• You will always be given these formulas in tests and exams; focus is on applying, not memorizing.
• Four main forms: sin(A+B), sin(A−B), cos(A+B), cos(A−B).

Expanding Using Compound Angle Formulas

• Identify the correct formula by matching the structure (sin/cos, plus/minus) to your expression.
• For example, cos(X−10) expands to cosX·cos10 + sinX·sin10.
• sin(20+y) expands to sin20·cosY + cos20·sinY.
• sin(50−W) expands to sin50·cosW − cos50·sinW.
• cos(T+5) expands to cosT·cos5 − sinT·sin5.

Using Special Angles

• Special angles are 30°, 45°, and 60°; break down larger angles using these (e.g., 75° = 45° + 30°).
• Expand using the relevant sum/difference formula and substitute special angle values (from calculator or special triangles).
• For example, cos75° = cos45°·cos30° − sin45°·sin30°, then substitute known values.
• When adding or subtracting fractions, keep common denominators.

Simplifying Compound Expressions

• To simplify, "reverse" the expansion: recognize patterns and convert back to a single sin or cos with a single angle.
• cos80°·cosW + sin80°·sinW simplifies to cos(80°−W).
• cos50°·cos10° − sin50°·sin10° simplifies to cos(50°+10°) = cos60°.
• sinT·cos20° − cosT·sin20° simplifies to sin(T−20°).
• sin10°·cos20° + cos10°·sin20° simplifies to sin(10°+20°) = sin30°.

Calculator and Special Triangle Notes

• For special angles, you may use calculator values or special triangle ratios; answers may look different but are equivalent.
• Always substitute and evaluate each part individually; do not enter the expanded form all at once.

Key Terms & Definitions

• Compound Angle Formula — Formulas expressing sin or cos of sum/difference as products or sums of sines and cosines.
• Special Angles — Angles with known exact values: 30°, 45°, 60°.
• Expand — To write a function of a sum/difference as a combination of products using the formula.
• Simplify — To recognize a combination of products as a single sin or cos of a sum/difference.

Action Items / Next Steps

• Practice identifying and applying the correct compound angle formula.
• Use calculator or special triangles to evaluate special angles when expanding.
• Complete homework on expanding and simplifying compound angles using given examples.