Mathematics Form 4 | Trigonometry (3) | Masomotele Free Learning

📋 Learning Outcomes

By the end of this lesson, you should be able to:

State and apply the sine rule
State and apply the cosine rule
Calculate the area of a triangle using trigonometry
Solve problems involving bearings and angles of elevation/depression
Apply trigonometry to 3D problems
Use the unit circle to define trig ratios for all angles
Solve trigonometric equations

🔍 Key Inquiry Question: How can we find unknown sides and angles in any triangle using trigonometric rules?

📐 The Sine Rule

Sine Rule Formula

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

Use when you know: 2 angles + 1 side (AAS/ASA) or 2 sides + non-included angle (SSA)

Example:

In △ABC: A=35°, B=65°, a=8 cm. Find b and C.

C = 180°−35°−65° = 80°

b/sin65° = 8/sin35° → b = 8×sin65°/sin35° = 12.65 cm

Ambiguous Case (SSA)

⚠️ When given 2 sides and a non-included angle, there may be 0, 1 or 2 solutions!

Example:

a=7, b=10, A=30°. Find B.

sinB = 10×sin30°/7 = 5/7 → B = 45.6° or B = 134.4°

Both solutions are valid (ambiguous case) → Two triangles possible

📐 The Cosine Rule

Cosine Rule Formulae

a² = b² + c² − 2bc·cos A

cos A = (b²+c²−a²) / 2bc

Use when: 3 sides known (SSS) or 2 sides + included angle (SAS)

Example 1 (SAS):

b=5, c=7, A=60°. Find a.

a² = 25+49−2(5)(7)cos60° = 74−35 = 39

a = √39 ≈ 6.24 cm

Example 2 (SSS):

a=6, b=8, c=10. Find angle A.

cosA = (64+100−36)/(2×8×10) = 128/160 = 0.8

A = cos⁻¹(0.8) ≈ 36.87°

📐 Area of a Triangle

Trigonometric Area Formula

Area = ½ab·sin C

where a, b are two sides and C is the included angle

Example:

a=8 cm, b=10 cm, C=45°. Find area.

Area = ½×8×10×sin45° = 40×0.7071 = 28.28 cm²

Heron's Formula (SSS)

Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2

Example:

a=5, b=6, c=7. s = 9. Area = √[9×4×3×2] = √216 ≈ 14.7 cm²

🧭 Bearings and Angles

Bearings

Bearings are measured clockwise from North (000° to 360°).

Example:

Ship sails from A on bearing 060° for 20 km to B. Then on bearing 150° for 15 km to C. Find AC and bearing of C from A.

Angle ABC = 150°−60° = 90° (or use geometry)

AC = √(20²+15²) = √625 = 25 km

Angles of Elevation and Depression

Example:

From a point 50m from base of a tower, angle of elevation of top is 40°. Find height.

h = 50×tan40° = 50×0.839 = 41.95 m

🔢 Trig Graphs and Equations

Solving Trig Equations

CAST Rule: All trig ratios positive in Q1. Sin+ in Q2. Tan+ in Q3. Cos+ in Q4.

Example: Solve sin x = 0.5 for 0° ≤ x ≤ 360°

Reference angle = sin⁻¹(0.5) = 30°

Sin positive in Q1 and Q2: x = 30° or x = 180°−30° = 30° or 150°

Example: Solve 2cos x − 1 = 0 for 0° ≤ x ≤ 360°

cos x = 0.5 → reference = 60°

Cos positive in Q1 and Q4: x = 60° or 300°

📊 Trig Identities

Key Identities

Identity	Formula
Pythagorean	sin²θ + cos²θ = 1
Quotient	tan θ = sin θ / cos θ
From Pythag	1 + tan²θ = sec²θ
From Pythag	1 + cot²θ = cosec²θ

Prove: (sin θ + cos θ)² = 1 + 2sin θ cos θ

LHS = sin²θ + 2sinθcosθ + cos²θ = (sin²θ+cos²θ) + 2sinθcosθ = 1 + 2sinθcosθ = RHS ✓

📦 3D Trigonometry

Problems in Three Dimensions

Example — Diagonal of a cuboid:

Cuboid 6m × 4m × 3m. Find length of space diagonal AG.

Face diagonal AC = √(6²+4²) = √52 = 2√13

Space diagonal AG = √(AC²+3²) = √(52+9) = √61 ≈ 7.81 m

Angle of elevation in 3D:

The angle of elevation of point P from point A on the ground where AP = 10 m and the height of P = 6 m:

tan θ = 6/10 → θ = tan⁻¹(0.6) ≈ 30.96°

✏️ Practice Problems

Problem 1

In △PQR: PQ=8, QR=10, angle Q=50°. Find PR and area of △PQR.

Solution:

PR² = 8²+10²−2(8)(10)cos50° = 64+100−102.8 = 61.2 → PR = 7.82

Area = ½×8×10×sin50° = 30.64 cm²

Problem 2

Solve: tan x = −1 for 0° ≤ x ≤ 360°

Solution:

Reference = 45°. Tan negative in Q2 and Q4.

x = 180°−45° = 135° or x = 360°−45° = 135° or 315°

📝 Quiz — Trigonometry (3)

Part A: Fill in the Blanks (5 marks)

1. The rule a/sinA = b/sinB is called the .

2. The rule a² = b²+c²−2bc·cosA is called the .

3. The angles of a triangle sum to degrees.

4. The area formula ½ab·sinC uses sides and their included angle.

5. The formula ½ab·sinC is used to find the of a triangle.

Part B: Short Answer (10 marks)

6. (4 marks) In △ABC: a=9, b=12, C=60°. Find c and the area of the triangle.

7. (3 marks) Solve: 2sin x = 1 for 0° ≤ x ≤ 360°.

8. (3 marks) From the top of a cliff 80m high, the angle of depression of a boat is 25°. Find the distance of the boat from the base of the cliff.