Topic A: Fundamental Trig Identities
Lessons 1–5 · Pythagorean, reciprocal, and quotient identities.
01–03 — Lessons 1–3
Pythagorean & Reciprocal Identities
- Pythagorean identity: sin²θ + cos²θ = 1
- Derived: 1 + tan²θ = sec²θ · 1 + cot²θ = csc²θ
- Reciprocal identities:
- csc θ = 1/sin θ · sec θ = 1/cos θ · cot θ = 1/tan θ
- Quotient identities: tan θ = sin θ/cos θ
If sin θ = 3/5, then cos²θ = 1 − 9/25 = 16/25 → cos θ = 4/5 (if θ in QI).
Practice — Lessons 1–3
Q1 · Multiple Choice
Which identity is equivalent to 1 − sin²θ?
Q2 · Short Answer
If cos θ = 5/13, find sin θ (assume θ in Quadrant I). Write as a fraction.
04–05 — Lessons 4–5
Verifying Trig Identities
To verify an identity, transform one side to match the other using known identities.
- Work on only ONE side at a time.
- Use algebraic skills: factor, combine fractions, multiply by conjugates.
- Rewrite everything in terms of sin and cos when stuck.
- An identity must hold for ALL values in the domain — one example isn't enough.
Verify: tan²θ + 1 = sec²θ. Left side: sin²θ/cos²θ + cos²θ/cos²θ = (sin²θ + cos²θ)/cos²θ = 1/cos²θ = sec²θ ✓
Practice — Lessons 4–5
Q3 · Multiple Choice
When verifying a trig identity, you should:
Q4 · Multiple Choice
Which is a valid first step to verify sin²θ/cos²θ = tan²θ?
Topic B: Solving Trigonometric Equations
Lessons 6–10 · Solving for angle values in a given domain.
06–10 — Lessons 6–10
Solving Trig Equations
Solve trig equations by isolating the trig function, then using inverse trig or the unit circle.
- Step 1: Isolate the trig function (e.g., get sin x = 0.5).
- Step 2: Find the reference angle using inverse trig.
- Step 3: Find all solutions in the given domain using quadrant analysis.
- On [0, 2π): sin x = k usually has 2 solutions; check the quadrant signs.
- Factor if multiple trig terms — use the zero product property.
sin x = √3/2, 0 ≤ x < 2π → x = π/3 and x = 2π/3 (both in QI and QII where sin > 0).
Practice — Lessons 6–10
Q5 · Multiple Choice
Solve 2cos x = 1 on [0, 2π). How many solutions?
Q6 · Short Answer
Solve sin x = 0 on [0, 2π). Give the solution between 0 and π exclusive.
Topic C: Sum/Difference & Double Angle Formulas
Lessons 11–16 · Advanced identities and their applications.
11–13 — Lessons 11–13
Sum & Difference Formulas
- sin(A ± B) = sin A cos B ± cos 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)
- Use to find exact values of angles like 75°, 15°, etc.
cos(75°) = cos(45°+30°) = cos45°cos30° − sin45°sin30°
= (√2/2)(√3/2) − (√2/2)(1/2) = (√6−√2)/4
Practice — Lessons 11–13
Q7 · Multiple Choice
sin(A + B) = sin A cos B + cos A sin B. If A = π/6, B = π/3, what is sin(π/2)?
Q8 · Multiple Choice
Which formula gives cos(A − B)?
14–16 — Lessons 14–16
Double Angle Formulas
- sin(2A) = 2 sin A cos A
- cos(2A) = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
- tan(2A) = 2tan A/(1 − tan²A)
- Derived from the sum formulas by setting B = A.
If sin A = 3/5 and cos A = 4/5:
sin(2A) = 2(3/5)(4/5) = 24/25.
Practice — Lessons 14–16
Q9 · Multiple Choice
If cos θ = 1/2, what is cos(2θ) using cos(2θ) = 2cos²θ − 1?
Q10 · Short Answer
If sin A = 4/5 and cos A = 3/5, find sin(2A).