Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Overview
Trigonometry rewards two habits: knowing the standard values cold, and drawing a correct, labelled right triangle for every heights-and-distances problem. Master those and this becomes a high-scoring chapter.
1. Trigonometric Ratios
sin θ = opposite / hypotenuse cosec θ = 1/sin θ
cos θ = adjacent / hypotenuse sec θ = 1/cos θ
tan θ = opposite / adjacent cot θ = 1/tan θ
2. Standard Angle Values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
3. Fundamental Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
These three (and their rearrangements) solve almost every "prove that" sum.
4. Heights and Distances
- Draw a labelled right triangle; mark the angle of elevation/depression.
- Choose the ratio that links the known side to the unknown (usually tan).
- Always write the unit in the answer.
Worked Examples
Example 1 — Prove an identity
Prove (1 − sin²θ) sec²θ = 1.
LHS = (1 − sin²θ) sec²θ = cos²θ × (1/cos²θ) = 1 = RHS
Example 2 — Evaluate with standard angles
Evaluate sin 30° cos 60° + cos 30° sin 60°.
= (1/2)(1/2) + (√3/2)(√3/2) = 1/4 + 3/4 = 1
(This equals sin(30° + 60°) = sin 90° = 1.)
Example 3 — Find a ratio
If tan θ = 3/4, find sin θ and cos θ.
opposite = 3, adjacent = 4 → hypotenuse = √(3² + 4²) = 5
sin θ = 3/5, cos θ = 4/5
Example 4 — Height (angle of elevation)
The angle of elevation of the top of a tower from a point 30√3 m away is 30°. Find the height.
tan 30° = h / (30√3)
1/√3 = h / (30√3) → h = 30√3 / √3 = 30 m
Example 5 — Distance (angle of depression)
From the top of a 60 m lighthouse, the angle of depression of a boat is 45°. Find the boat's distance from the base.
tan 45° = 60 / d → 1 = 60/d → d = 60 m
Common Mistakes
- Mixing up opposite and adjacent sides.
- Forgetting the standard values (especially 30° and 60°).
- Starting an identity proof from both sides at once — work from one side only.
- Measuring the angle from the object instead of the horizontal.
Exam Tips & Revision Checklist
- Memorise the standard-value table and the three identities.
- For word problems: diagram first, then pick the ratio.
- Keep surds (√3, √2) exact unless a decimal is asked.
- Practised: ratios, standard values, identity proofs, elevation & depression.
Frequently Asked Questions
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Their reciprocals are cosec, sec, and cot.