Ch 8Introduction to Trigonometry

Chapter 8: Introduction to Trigonometry

1. What is Trigonometry?

Trigonometry (from Greek: tri = three, gon = sides, metron = measure) is the study of relationships between the sides and angles of a triangle. In this chapter, all ratios are defined for an acute angle of a right triangle, and the study is motivated for angles from 0° to 90°.

2. Trigonometric Ratios

In right triangle ABC, right angled at B, for the acute angle A:

• Side opposite to angle A = BC (perpendicular, P)
• Side adjacent to angle A = AB (base, B)
• Hypotenuse = AC (H) — the side opposite the right angle, always the longest side

3. Key Facts About the Ratios

• The trigonometric ratios of an angle depend only on the angle, not on the size of the triangle (this follows from similarity of triangles: any two right triangles with the same acute angle are similar by AA).
• Since the hypotenuse is the longest side, sin A ≤ 1 and cos A ≤ 1 always. Hence cosec A ≥ 1 and sec A ≥ 1.
• tan A and cot A can take any positive value; tan A is not defined at 90° and cot A is not defined at 0°.
• "sin A" is one symbol; sin cannot be separated from A. Notation: sin²A means (sin A)², but never write sin A².
• If one trigonometric ratio of an acute angle is known, all other ratios can be determined (draw a right triangle, use the Pythagoras result to find the third side, then read off the ratios).

4. Trigonometric Ratios of Standard Angles

• As A increases from 0° to 90°, sin A increases from 0 to 1, cos A decreases from 1 to 0, and tan A increases from 0 without bound.
• Derivations: for 45° use an isosceles right triangle with legs 1, 1 and hypotenuse √2; for 30° and 60° use an equilateral triangle of side 2a with an altitude (halves: a and √3a).

5. Trigonometric Identities

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle involved. In the rationalised syllabus, only one fundamental identity and its derived forms are required:

Proof of the fundamental identity: In right triangle ABC right angled at B, by the Pythagoras result, AB² + BC² = AC². Dividing every term by AC²: (AB/AC)² + (BC/AC)² = 1, i.e. cos²A + sin²A = 1.

Useful rearrangements (use freely in proofs):

• sin²A = 1 − cos²A and cos²A = 1 − sin²A
• sec²A − tan²A = 1, so (sec A − tan A)(sec A + tan A) = 1
• cosec²A − cot²A = 1, so (cosec A − cot A)(cosec A + cot A) = 1

6. Method for "Prove the Identity" Questions

1. Start from the more complicated side (usually LHS).
2. Convert everything into sin and cos — this works in almost every board question.
3. Take LCM, simplify, and use sin²A + cos²A = 1 (or a rearrangement) to reduce.
4. Rationalise by multiplying with a conjugate, e.g. multiply (1 − cos A) expressions by (1 + cos A), when square roots or 1 ± forms appear.
5. End with "= RHS. Hence proved."

7. Common Board Question Patterns

• Given one ratio (e.g. tan A = 3/4), find all other ratios or evaluate an expression (2/3 marks).
• Evaluate expressions using the standard-angle table, e.g. 2 tan²45° + cos²30° − sin²60° (2 marks).
• Prove identities such as (cosec A − cot A)² = (1 − cos A)/(1 + cos A) (3 marks).
• If sin A + cos A or tan A + cot A type conditions are given, find the value of a symmetric expression (3 marks).
• Assertion-Reason and MCQs on maximum/minimum values, "not defined" values, and increasing/decreasing behaviour (1 mark).