Topic 12: Trigonometry

🟢 Lite — Quick Review (1h–1d)

Trigonometry is the study of the relationships between the angles and sides of triangles. The three primary trigonometric ratios—sine, cosine, and tangent—are defined for acute angles in right-angled triangles. For an angle θ, with reference to a right-angled triangle: sine (sin θ) = opposite/hypotenuse, cosine (cos θ) = adjacent/hypotenuse, and tangent (tan θ) = opposite/adjacent. These ratios are constant for a given angle regardless of the size of the triangle, which makes them powerful tools for solving geometric problems.

The values of these ratios for key angles must be committed to memory, as they appear frequently in the WASSCE examination. For 30°, sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3. For 45°, sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1. For 60°, sin 60° = √3/2, cos 60° = ½, tan 60° = √3. The relationship sin²θ + cos²θ = 1 (derived from Pythagoras) is always true and frequently useful for verification.

Key Facts:

• SOHCAHTOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent
• sin²θ + cos²θ = 1 (fundamental identity)
• tan θ = sin θ / cos θ
• Angles of elevation and depression are measured from the horizontal
• The amplitude of sin and cos graphs is 1; period is 360° (or 2π radians)
• Sine and cosine values are always between -1 and 1

⚡ Exam Tip: In WASSCE word problems involving heights and distances, always draw a right-angled triangle. Label the known sides, identify the relevant trigonometric ratio, and form an equation.

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🟡 Standard — Regular Study (2d–2mo)

Finding Angles Using Inverse Trigonometry

If sin θ = 0.5, find θ (0° ≤ θ ≤ 90°) θ = sin⁻¹(0.5) = 30°

If tan θ = 1, find θ θ = tan⁻¹(1) = 45°

Solving Right-Angled Triangle Problems

A ladder 10 metres long leans against a wall making an angle of 65° with the ground. Find: (a) How far the foot of the ladder is from the wall (b) How high up the wall the ladder reaches

(a) Adjacent to angle = cos 65° = adjacent/10 Adjacent = 10 × cos 65° = 10 × 0.4226 = 4.23 m

(b) Opposite to angle = sin 65° = opposite/10 Opposite = 10 × sin 65° = 10 × 0.9063 = 9.06 m

Angles of Elevation and Depression

The angle of elevation is the angle between the horizontal and the line of sight looking UP to an object. The angle of depression is the angle between the horizontal and the line of sight looking DOWN to an object.

A man standing 40 metres from a tree observes the top at an angle of elevation of 35°. Find the height of the tree.

tan 35° = height/40 height = 40 × tan 35° = 40 × 0.7002 = 28.0 m

Trigonometric Identities

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

LHS = sin²θ + 2 sin θ cos θ + cos²θ = (sin²θ + cos²θ) + 2 sin θ cos θ = 1 + 2 sin θ cos θ = RHS ✓

The Sine Rule

For any triangle ABC with sides a, b, c opposite angles A, B, C: a/sin A = b/sin B = c/sin C

Find side b if a = 8 cm, angle A = 40°, and angle B = 65°.

b/sin 65° = 8/sin 40° b = 8 × sin 65°/sin 40° b = 8 × 0.9063/0.6428 = 11.3 cm

The Cosine Rule

For triangle ABC: a² = b² + c² - 2bc cos A

Find angle A if a = 7, b = 5, c = 6. 49 = 25 + 36 - 2(5)(6) cos A 49 = 61 - 60 cos A 60 cos A = 12 cos A = 0.2 A = cos⁻¹(0.2) ≈ 78.5°

Comparison Table: Sine vs Cosine Rule

Property	Sine Rule	Cosine Rule
Used when	Two angles and one side known, or two sides and an angle opposite one of them	Two sides and the included angle known, or all three sides known
Formula	a/sin A = b/sin B = c/sin C	a² = b² + c² - 2bc cos A
Finds	Any side or angle	Any angle (with three sides) or any side (with two sides and included angle)

Common Mistakes to Avoid:

1. Using the wrong trigonometric ratio (SOHCAHTOA confusion)
2. Confusing which side is opposite, adjacent, or the hypotenuse
3. Using degrees instead of radians or vice versa when required
4. Forgetting that sin, cos, and tan values can be greater than 1 (only sin and cos are bounded by -1 and 1)
5. Mixing up angle of elevation and angle of depression

Problem-Solving Strategy:

1. Draw a clear, labelled diagram
2. Identify the right angle and label the triangle
3. Determine which sides are known and which is required
4. Select the appropriate trigonometric ratio
5. Form an equation and solve
6. Check that your answer is sensible in the context of the problem

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🔴 Extended — Deep Study (3mo+)

Trigonometric Graphs

The graph of y = sin θ has:

• Amplitude = 1
• Period = 360° (or 2π radians)
• Maximum at 90° (value = 1)
• Minimum at 270° (value = -1)
• Y-intercept at 0° = 0

The graph of y = cos θ has:

• Amplitude = 1
• Period = 360°
• Maximum at 0° (value = 1)
• Minimum at 180° (value = -1)
• Y-intercept at 0° = 1

The graph of y = tan θ has:

• No amplitude (unbounded)
• Period = 180°
• Asymptotes at 90° and 270°
• Zeroes at 0°, 180°, 360°

Radians

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. 180° = π radians, so 1 radian ≈ 57.3°

Conversion:

• To convert degrees to radians: multiply by π/180
• To convert radians to degrees: multiply by 180/π

Area of a Triangle Using Trigonometry

Area = ½ab sin C, where a and b are two sides and C is the included angle.

For triangle with sides 7 cm and 10 cm, and included angle 55°: Area = ½ × 7 × 10 × sin 55° = 35 × 0.8192 = 28.67 cm²

The Ambiguous Case — Sine Rule

When using the sine rule to find an angle given sin θ = k (where 0 < k < 1), there are always two possible angles: θ and (180° - θ). Both may be valid in triangle problems. For example, if sin θ = 0.5, then θ could be 30° or 150°. The context of the problem determines which (or both) solutions are valid.

Compound Angle 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)

Using sin(A + B): sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

Double Angle Formulas

sin 2A = 2 sin A cos A cos 2A = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A tan 2A = 2 tan A/(1 - tan²A)

WASSCE Examination Patterns:

The WASSCE quantitative reasoning paper typically includes:

1. Finding sides using SOHCAHTOA (Objective)
2. Finding angles using inverse trigonometric functions (Objective)
3. Angles of elevation and depression problems (Theory)
4. Sine rule and cosine rule applications (Theory)
5. Trigonometric identities (Theory)

Exact Values Table:

Angle	sin	cos	tan
0°	0	1	0
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	undefined

⚡ Pro Exam Tip: In WASSCE, always check whether your answer should be in degrees or radians based on the question. When using the sine rule for angles, consider the ambiguous case—if asked for “an angle” rather than “the angle,” there may be two possible answers.

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