Topic 6: Trigonometry

Trigonometry Notes | Visual Learner Guide

📐 Trigonometry Mastery

Right-angled triangles · Exact values · Graphs · Sine & Cosine rules · 3D problems

📐 Pythagoras’ theorem

🔺 For right-angled triangles only – the square of the hypotenuse equals sum of squares of the other two sides.

a² + b² = c² (c = hypotenuse)

✧ Find hypotenuse: c = √(a² + b²) | ✧ Find shorter side: a = √(c² – b²)

💡 Always identify the hypotenuse first — it’s opposite the right angle and the longest side.

📐 Right-angled triangles: sin, cos, tan

sin θ = opposite / hypotenuse | cos θ = adjacent / hypotenuse | tan θ = opposite / adjacent

SOH CAH TOA – use to find missing sides or angles (acute angles). For an angle: θ = sin⁻¹(opp/hyp) etc.

🎯 Find side → multiply or rearrange formula

🎯 Find angle → inverse trig functions

✔️ Also: solve 2D geometry problems by identifying right triangles in shapes.

✨ Perpendicular distance from a point to a line = shortest distance (forms a right angle).

⛰️ Angles of elevation & depression

Elevation = angle above horizontal (looking up). Depression = angle below horizontal (looking down).

🔁 Angle of depression from A to B = Angle of elevation from B to A (alternate interior angles).

Always draw a horizontal line at eye level — creates right‑angled triangle with height and distance.

⭐ Exact trigonometric values

Memorise these for quick, exact answers (no calculator for special angles).

Angle (x)	0°	30°	45°	60°	90°
sin x	0	12	√22	√32	1
cos x	1	√32	√22	12	0
tan x	0	1√3	1	√3	undefined

🧠 Mnemonic: sin values go √02, √12, √22, √32, √42 → 0, ½, √2/2, √3/2, 1. Cos is reverse.

📈 Graphs of sin, cos, tan (0° – 360°)

y = sin x : starts at 0 → max 1 at 90° → 0 at 180° → min -1 at 270° → returns to 0 at 360°.
y = cos x : starts at 1 → 0 at 90° → -1 at 180° → 0 at 270° → back to 1 at 360°.
y = tan x : period 180°, asymptotes at 90° and 270° (undefined), crosses through 0°, 180°, 360°.

📌 Know intercepts, maxima/minima, asymptotes for tan. CAST diagram helps solve equations.

✏️ Solving trig equations (0° ≤ x ≤ 360°): Find reference angle, use graph / CAST.
Example: sin x = 12 → x = 30° and 150°. Cos positive in Q1+Q4, tan positive in Q1+Q3.

🧭 CAST: Q1 All +, Q2 Sin +, Q3 Tan +, Q4 Cos +.

🌊 Sine Rule (non‑right triangles)

a / sin A = b / sin B = c / sin C

✔️ Use when you know: two angles + one side (AAS, ASA) or two sides + non‑included angle (ASS ambiguous case).

To find a side: a = (b × sin A)/ sin B | To find an angle: sin A = (a × sin B)/ b (check for two possible angles if acute/obtuse).

⚠️ Label carefully: side a is opposite angle A, side b opposite angle B.

🌀 Cosine Rule

For finding a side (SAS):

a² = b² + c² – 2bc·cos A

For finding an angle (SSS):

cos A = (b² + c² – a²) / (2bc)

Perfect when you know two sides and the included angle, or all three sides.

📏 Area of any triangle (non‑right)

Area = 12 · a · b · sin C

Where a and b are two sides, and C is the included angle between them. Works for all triangles, even right‑angled (sin 90° = 1).

✍️ Use when you have two side lengths and the angle between them.

🧊 3D Pythagoras & trigonometry

Key idea: find right‑angled triangles hidden in 3D solids (cubes, cuboids, pyramids).

Angle between a line and a plane:
1. Project the line onto the plane (drop perpendicular).
2. The required angle = angle between the original line and its projection.

📐 Sketch the 2D triangle separately → apply Pythagoras or trig.

Example: space diagonal of a cuboid: length = √(l² + w² + h²). Always identify the right‑angled triangle within the solid.

⭐ For angle between line & plane: the perpendicular distance matters — same logic as 2D shortest distance.

📌 Quick reference – when to use which rule

✔ Right‑angled triangle
Pythagoras / SOH CAH TOA

✔ Two angles + one side
Sine rule

✔ Two sides + included angle
Cosine rule (find side)

✔ Three sides (SSS)
Cosine rule (find angle)

✔ Two sides + non‑included
Sine rule (check ambiguous)

✔ Area given 2 sides & incl angle
Area = 12 ab sin C

🎯 Perpendicular distance reminder: shortest distance from a point to a line = perpendicular → forms a right angle.

🧩 Solving equations (0°–360°) & CAST diagram

Solve sin x = k, cos x = k, tan x = k → find reference angle α = sin⁻¹|k| (or cos⁻¹, tan⁻¹). Then use quadrant rules:

✦ sin positive: Q1 & Q2 → x = α, 180° – α
✦ cos positive: Q1 & Q4 → x = α, 360° – α
✦ tan positive: Q1 & Q3 → x = α, 180° + α

For negative values, find where function is negative (e.g., sin negative → Q3, Q4).

📈 Graph sketching: y = sin x and y = cos x have amplitude 1, period 360°. y = tan x repeats every 180°, vertical asymptotes at 90°, 270°.

✨ Exact values – quick reference

🔹 sin 30° = 12 🔹 sin 45° = √22 🔹 sin 60° = √32
🔹 cos 30° = √32 🔹 cos 45° = √22 🔹 cos 60° = 12
🔹 tan 30° = 1√3 🔹 tan 45° = 1 🔹 tan 60° = √3

💡 These appear frequently in exam problems without calculators — memorise them well.

Practice Papers

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