Concepts & Worked Examples
Study each unit carefully. Memorise the key formulas before attempting the problems.
Unit 1
Angle Measure & the Unit Circle
Key Formulas
- Arc length: s = r · θ (θ in radians)
- Sector area: A = ½ · r² · θ
- Degree ↔ Radian: θ(rad) = θ(deg) · (π/180)
- Linear speed: v = r · ω, Angular speed: ω = θ / t
Unit Circle — Exact Values
- sin 0=0 · sin(π/6)=½ · sin(π/4)=√2/2 · sin(π/3)=√3/2 · sin(π/2)=1
- cos 0=1 · cos(π/6)=√3/2 · cos(π/4)=√2/2 · cos(π/3)=½ · cos(π/2)=0
- tan = sin / cos (undefined where cos = 0)
- ASTC rule — All · Sin · Tan · Cos positive in Q1, Q2, Q3, Q4
Worked Example
Find the exact value of tan(7π/6).
Solution: tan(7π/6) = tan(π + π/6) = tan(π/6) = √3/3
Unit 2
Graphs of Trigonometric Functions
Transformations of y = A·sin(Bx + C) + D
- Amplitude = |A|
- Period = 2π / |B|
- Phase shift = −C/B (positive → right; negative → left)
- Vertical shift = D
- sin is ODD: sin(−x) = −sin x | cos is EVEN: cos(−x) = cos x
Worked Example
State amplitude, period, and phase shift of y = 3sin(2x − π/4).
Solution: Amplitude = 3, Period = π, Phase shift = π/8 to the right
Unit 3
Trigonometric Identities
Fundamental Identities
- Pythagorean: sin²x + cos²x = 1
- 1 + tan²x = sec²x
- 1 + cot²x = csc²x
- Reciprocal: csc x = 1/sin x | sec x = 1/cos x | cot x = cos x/sin x
- Quotient: tan x = sin x / cos x
Sum, Difference & Double-Angle
- 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)
- sin(2A) = 2 sin A cos A
- cos(2A) = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
- tan(2A) = 2 tan A / (1 − tan²A)
Half-Angle & Product-to-Sum
- sin(A/2) = ±√((1 − cos A)/2)
- cos(A/2) = ±√((1 + cos A)/2)
- tan(A/2) = sin A/(1 + cos A) = (1 − cos A)/sin A
- sin A cos B = ½[sin(A+B) + sin(A−B)]
- cos A cos B = ½[cos(A−B) + cos(A+B)]
Worked Example
Evaluate sin(75°) exactly using the sum formula.
Solution: sin(45°+30°) = sin45°cos30°+cos45°sin30° = (√6+√2)/4
Unit 4
Inverse Trigonometric Functions
Domains & Ranges
- arcsin x: domain [−1, 1], range [−π/2, π/2]
- arccos x: domain [−1, 1], range [0, π]
- arctan x: domain ℝ, range (−π/2, π/2)
- Composition: sin(arcsin x) = x for x ∈ [−1, 1]
- arcsin(sin x) = x ONLY if x ∈ [−π/2, π/2]
Worked Example
Evaluate cos(arctan(3/4)).
Solution: Draw right △: opp=3, adj=4, hyp=5. cos(arctan(3/4)) = 4/5
Unit 5
Solving Trigonometric Equations
Strategy
- 1. Isolate the trig function
- 2. Find reference angle using inverse trig
- 3. Use ASTC / quadrant rules → all solutions in [0, 2π)
- 4. Add period multiples for general solution
- 5. Check domain restrictions
Worked Example
Solve 2sin²x − sin x − 1 = 0 on [0, 2π).
Solution: (2sin x+1)(sin x−1)=0 → x = 7π/6, 11π/6, π/2
Unit 6
Law of Sines & Cosines
Laws
- Law of Sines: a/sin A = b/sin B = c/sin C
- Law of Cosines: a² = b² + c² − 2bc · cos A
- cos A = (b²+c²−a²) / (2bc)
- Area: K = ½ab sin C
- Heron: K = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
Worked Example
In △ABC: a=7, b=5, C=60°. Find c.
Solution: c² = 49+25−70cos60° = 39 → c = √39 ≈ 6.24
Unit 7
Polar Coordinates & Complex Numbers
Key Conversions & DeMoivre's Theorem
- Polar → Rect: x = r cos θ, y = r sin θ
- Rect → Polar: r = √(x²+y²), θ = arctan(y/x) (adjust quadrant!)
- Trig form: z = r(cos θ + i sin θ)
- DeMoivre: z^n = r^n (cos(nθ) + i sin(nθ))
- nth roots: θ_k = (θ + 2kπ)/n, k = 0, 1, …, n−1
Worked Example
Write z = 1 + i in trig form, then find z⁴.
Solution: r=√2, θ=π/4. z⁴ = (√2)⁴(cos π + i sin π) = −4
Practice Problems
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Answer Key & Full Solutions
Complete solutions for all 20 problems. Study the method, not just the answer.