Trigonometry | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Understand trigonometric ratios (sin, cos, tan) in right-angled trianglesFahami nisbah trigonometri (sin, cos, tan) dalam segitiga bersudut tegak

Apply ASTC rule for trigonometric ratios in different quadrantsGunakan peraturan ASTC untuk nisbah trigonometri dalam kuadran berbeza

Use sine and cosine rules to solve trianglesGunakan peraturan sinus dan kosinus untuk menyelesaikan segitiga

Calculate area of triangle using trigonometric formulasKira luas segitiga menggunakan formula trigonometri

Sketch graphs of trigonometric functionsLukis graf fungsi trigonometri

📖 Key Concepts📖 Konsep Utama

Trigonometric Ratios (Form 3)Nisbah Trigonometri (Tingkatan 3)

In a right-angled triangle, the three basic ratios relate angles to side lengths.Dalam segitiga bersudut tegak, tiga nisbah asas menghubungkan sudut dengan panjang sisi.

SOH-CAH-TOASOH-CAH-TOA
sin θ = opp/hyp · cos θ = adj/hyp · tan θ = opp/adj

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
SOH-CAH-TOA memory aid

ASTC Rule (Form 4)Peraturan ASTC (Tingkatan 4)

Determines the sign of trigonometric ratios in each quadrant of the Cartesian plane.Tentukan tanda nisbah trigonometri dalam setiap kuadran pada satah Cartesan.

Q1: All positive (A) | Q2: Sine positive (S)
Q3: Tan positive (T) | Q4: Cos positive (C)
Reference angle: always measured from x-axis

Sine Rule (Form 4)Peraturan Sinus (Tingkatan 4)

Used for any triangle when you know an angle-side pair.Digunakan untuk mana-mana segitiga apabila anda tahu pasangan sudut-sisi.

a / sin A = b / sin B = c / sin C
Use when: two angles and one side known, or two sides and one opposite angle known

Cosine Rule (Form 5)Peraturan Kosinus (Tingkatan 5)

Relates all three sides and one angle. Use when sine rule doesn't apply.Hubungkan ketiga-tiga sisi dan satu sudut. Gunakan apabila peraturan sinus tidak boleh digunakan.

a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) / 2bc
Use when: two sides and included angle, or all three sides known

Area of Triangle (Form 4)Luas Segitiga (Tingkatan 4)

Calculate area using two sides and the included angle.Kira luas menggunakan dua sisi dan sudut included.

Area = ½ × a × b × sin C
Works for ANY triangle (not just right-angled)

Angles of Elevation & Depression (Form 4)Sudut Dongak & Tunduk (Tingkatan 4)

Elevation = looking UP from horizontal. Depression = looking DOWN from horizontal.Dongak = memandang KE ATAS dari ufuk. Tunduk = memandang KE BAWAH dari ufuk.

🔴 Angle of Elevation (α) = looking UP from observer | 🟡 Angle of Depression (β) = looking DOWN from top🔴 Sudut Dongak (α) = memandang KE ATAS | 🟡 Sudut Tunduk (β) = memandang KE BAWAH

✏️ Worked Examples✏️ Contoh Penyelesaian

In right-angled triangle ABC, AB = 5 cm, angle A = 30°. Find BC.Dalam segitiga bersudut tegak ABC, AB = 5 cm, sudut A = 30°. Cari BC.

Identify: opposite to A is BC, adjacent is ACKenal pasti: bertentangan dengan A ialah BC, bersisian ialah AC

tan 30° = BC / AB = BC / 5

BC = 5 × tan 30° = 5 × (1/√3) ≈ 2.887

BC ≈ 2.89 cm (3 s.f.)

Triangle ABC: AB = 7 cm, angle A = 50°, angle B = 60°. Find BC.Segitiga ABC: AB = 7 cm, sudut A = 50°, sudut B = 60°. Cari BC.

Find angle C: 180° − 50° − 60° = 70°Cari sudut C: 180° − 50° − 60° = 70°

Sine rule: AB / sin C = BC / sin APeraturan sinus: AB / sin C = BC / sin A

BC = 7 × sin 50° / sin 70° ≈ 7 × 0.7660 / 0.9397 ≈ 5.68

BC ≈ 5.68 cm

Find side a when b = 5, c = 7, angle A = 40°.Cari sisi a apabila b = 5, c = 7, sudut A = 40°.

Cosine rule: a² = b² + c² − 2bc cos APeraturan kosinus: a² = b² + c² − 2bc cos A

a² = 25 + 49 − 2 × 5 × 7 × cos 40° = 74 − 53.62 = 20.38

a = √20.38 ≈ 4.51

a ≈ 4.51 cm

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Using the wrong trig ratio for the given angleMenggunakan nisbah trig yang salah untuk sudut yang diberi

Always identify opposite, adjacent and hypotenuse relative to the angle. Draw the triangle and label sides before choosing sin, cos or tan.Sentiasa kenal pasti bertentangan, bersisian dan hipotenus berdasarkan sudut. Lukis segitiga dan label sisi sebelum memilih sin, cos atau tan.

❌ Forgetting ASTC rule for angles greater than 90°Lupa peraturan ASTC untuk sudut lebih daripada 90°

Angles in Q2–Q4 need reference angles. Remember: Q2 = 180°−θ, Q3 = 180°+θ, Q4 = 360°−θ. Always check the sign!Sudut dalam Q2–Q4 perlu sudut rujukan. Ingat: Q2 = 180°−θ, Q3 = 180°+θ, Q4 = 360°−θ. Sentiasa semak tanda!

❌ Wrong pairing in sine rulePemasangan salah dalam peraturan sinus

a/sin A = b/sin B — side a pairs with angle A, side b pairs with angle B. Don't mix them up! Each side must match its opposite angle.a/sin A = b/sin B — sisi a berpasangan dengan sudut A, sisi b berpasangan dengan sudut B. Jangan campurkan! Setiap sisi mesti sepadan dengan sudut bertentangannya.

📝 SPM Practice📝 Amalan SPM

Paper 1 — Multiple ChoiceKertas 1 — Pilihan Berganda

In a right-angled triangle, sin θ = 3/5. What is cos θ?Dalam segitiga bersudut tegak, sin θ = 3/5. Apakah cos θ?

4/5

5/4

3/4

Using ASTC, tan θ is positive in which quadrants?Menggunakan ASTC, tan θ positif di kuadran manakah?

1 and 3

2 and 4

1 and 2

3 and 4

Sine rule: If a = 5, A = 30°, B = 60°, find b.Peraturan sinus: Jika a = 5, A = 30°, B = 60°, cari b.

2.5

8.66

5.0

4.33

Cosine rule: Find angle C if a = 6, b = 8, c = 10.Peraturan kosinus: Cari sudut C jika a = 6, b = 8, c = 10.

30°

45°

60°

90°

Area of triangle with sides 5, 7 and included angle 40°.Luas segitiga dengan sisi 5, 7 dan sudut included 40°.

11.25

17.5

10.0

15.0

Paper 2 — Structured QuestionsKertas 2 — Soalan Berstruktur

5 marks5 markah

Triangle PQR, PQ = 8 cm, QR = 10 cm, angle P = 45°.
(a) Find PR using cosine rule. [3 marks]
(b) Find angle Q. [2 marks]Segitiga PQR, PQ = 8 cm, QR = 10 cm, sudut P = 45°.
(a) Cari PR menggunakan peraturan kosinus. [3 markah]
(b) Cari sudut Q. [2 markah]

Solution (a)Penyelesaian (a)

PR² = PQ² + QR² − 2 × PQ × QR × cos P

PR² = 64 + 100 − 2 × 8 × 10 × cos 45°

PR² = 164 − 160 × 0.7071 = 164 − 113.14 = 50.86

PR ≈ 7.13 cm [3 marks3 markah]

Solution (b)Penyelesaian (b)

Sine rule: sin Q / QR = sin P / PRPeraturan sinus: sin Q / QR = sin P / PR

sin Q = 10 × sin 45° / 7.13 ≈ 0.9913

Angle Q ≈ 53.1° [2 marks2 markah]

4 marks4 markah

Find the area of triangle with sides 6 cm, 8 cm and included angle 50°.Cari luas segitiga dengan sisi 6 cm, 8 cm dan sudut included 50°.

SolutionPenyelesaian

Area = ½ × a × b × sin CLuas = ½ × a × b × sin C

Area = ½ × 6 × 8 × sin 50° = 24 × 0.7660

Area ≈ 18.4 cm² [4 marks4 markah]

📋 Formula Summary📋 Ringkasan Formula

Trigonometry:Trigonometri:

Basic Ratios:Nisbah Asas:
sin θ = opp/hyp | cos θ = adj/hyp | tan θ = opp/adj

ASTC Rule:Peraturan ASTC:
Q1: All + | Q2: Sine + | Q3: Tan + | Q4: Cos +

Sine Rule:Peraturan Sinus: a/sin A = b/sin B = c/sin C

Cosine Rule:Peraturan Kosinus: a² = b² + c² − 2bc cos A

Area:Luas: ½ab sin C