Introduction to Trigonometry — Study Notes

1. Introduction

Trigonometry is the study of relationships between the sides and angles of a triangle, derived from the Greek words "trigonon" (triangle) and "metron" (measure). This introductory chapter focuses on right-angled triangles and defines the fundamental trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. These ratios connect the acute angles of a right triangle to the ratios of its sides.

The chapter defines the trigonometric ratios for the angle theta in a right triangle, establishes the important identities connecting these ratios, and computes the exact values of the ratios for the standard angles 0, 30, 45, 60, and 90 degrees. A good understanding of the angle-to-ratio correspondence prepares students for the applications chapter that follows.

Trigonometry has enormous practical importance. It is used in astronomy to measure distances to stars, in navigation and surveying to compute heights and distances, in engineering and architecture, and in physics. The trigonometric identities and ratios learned here are the permanent foundation for all higher trigonometry, calculus, and geometry.

2. Trigonometric Ratios

Consider a right triangle ABC with the right angle at B. Let angle C = theta. Then relative to angle theta: - The side opposite angle C is AB (the perpendicular), call it p. - The side adjacent to angle C is BC (the base), call it b. - The hypotenuse is AC, call it h.

The six trigonometric ratios are defined as: sin theta = p/h = opposite/hypotenuse cos theta = b/h = adjacent/hypotenuse tan theta = p/b = opposite/adjacent

cosec theta = h/p = 1/sin theta sec theta = h/b = 1/cos theta cot theta = b/p = 1/tan theta

A useful mnemonic is "Pandit Badri Prasad Har Har Bole": P/B for tan, B/H for cos, P/H for sin.

3. Trigonometric Ratios of Some Specific Angles

The exact values of the trigonometric ratios for the standard angles are essential to memorise:

For angle 30 degrees: sin = 1/2, cos = sqrt(3)/2, tan = 1/sqrt(3) For angle 45 degrees: sin = 1/sqrt(2), cos = 1/sqrt(2), tan = 1 For angle 60 degrees: sin = sqrt(3)/2, cos = 1/2, tan = sqrt(3) For angle 0 degrees: sin = 0, cos = 1, tan = 0 For angle 90 degrees: sin = 1, cos = 0, tan is not defined

These values are derived from right triangles, including the 30-60-90 triangle and the isosceles right triangle, and they appear repeatedly in evaluation questions.

Table of Standard Values

| Angle | 0 | 30 | 45 | 60 | 90 | | sin | 0 | 1/2 | 1/sqrt(2) | sqrt(3)/2 | 1 | | cos | 1 | sqrt(3)/2 | 1/sqrt(2) | 1/2 | 0 | | tan | 0 | 1/sqrt(3) | 1 | sqrt(3) | not defined |

4. Trigonometric Identities

An identity is an equation true for all values of the variable. The three fundamental trigonometric identities are: sin^2 theta + cos^2 theta = 1 1 + tan^2 theta = sec^2 theta 1 + cot^2 theta = cosec^2 theta

From these, we can derive all others: - sin^2 theta = 1 - cos^2 theta - tan theta = sin theta/cos theta - cot theta = cos theta/sin theta

Example: Prove that (1 - sin theta)(1 + sin theta) = cos^2 theta. LHS = 1 - sin^2 theta = cos^2 theta = RHS.

These identities are used both to evaluate expressions and to prove more complicated identities, a standard board examination task.

5. Proving Trigonometric Identities

The standard strategy for proving identities:

1. Start with the more complicated side of the identity.
2. Express everything in terms of sin and cos if necessary.
3. Use the fundamental identities sin^2 + cos^2 = 1, tan = sin/cos, and the reciprocal relations.
4. Simplify step by step until both sides match.

Example: Prove that sin theta/cos theta = tan theta using the definitions. This is immediate from tan theta = p/b divided by the sin and cos definitions, and it is the most-used conversion in identity proofs.

When both sides look complicated, converting all functions into sin and cos is almost always a reliable starting point.

6. Complementary Angle Relations

For two angles that add to 90 degrees, the trigonometric ratios satisfy: sin(90 - theta) = cos theta cos(90 - theta) = sin theta tan(90 - theta) = cot theta cosec(90 - theta) = sec theta sec(90 - theta) = cosec theta cot(90 - theta) = tan theta

Example: sin 30 = cos 60 = 1/2, and tan 45 = cot 45 = 1. These relations help evaluate expressions like sin 35/cos 55 = sin 35/sin 35 = 1.

These co-function relations simplify many expressions by converting angles into matching functions.

Quick Revision Tables

Table 1: Trigonometric Ratios for Standard Angles

Table 2: Fundamental Identities

Mind Map

Important Diagrams (SVG)

Diagram 1: Right Triangle with Trigonometric Ratios

Diagram 2: Standard Angle Triangle (30-60-90)

Common Mistakes

1. Writing sin as opposite/adjacent or cos as opposite/hypotenuse; the correct pairs are sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
2. Confusing reciprocals: cosec is the reciprocal of sin, not of cos; sec is the reciprocal of cos, not of sin.
3. Forgetting that tan 90 degrees is not defined, and writing it as infinite or a large number.
4. In identities, writing sin^2 + cos^2 = 1 and then mistakenly using 1 - sin^2 = cos^2 as 1 - sin^2 = -cos^2.
5. Mixing degrees and radians, or using approximate values instead of the exact surd values for standard angles.
6. Forgetting the co-function relations: sin(90 - theta) equals cos theta, not sin theta.
7. Dividing both sides by a trigonometric function that could be zero, which invalidates the identity proof.

Exam Tips

1. Draw and label the right triangle for each problem, clearly marking the angle, opposite, adjacent, and hypotenuse.
2. Memorise the standard value table in both directions: from angle to ratio and from ratio to angle.
3. In identity proofs, convert everything to sin and cos as a first move; this resolves most problems.
4. Use sin^2 theta + cos^2 theta = 1 repeatedly by rearranging it in whichever form the question needs.
5. For evaluation questions, first replace complementary angles, e.g., cos 55 = sin 35, to simplify expressions.
6. Always rationalise the denominator when leaving answers like 1/sqrt(2) or 1/sqrt(3).
7. Practise proving identities with both sides simplified independently, as this is a guaranteed exam question.

Conclusion

Introduction to Trigonometry opens the door to one of the most applicable branches of mathematics. The six trigonometric ratios, their values at the standard angles, and the fundamental identities form a compact but powerful set of tools for solving problems involving triangles and angles. This chapter not only prepares students for the applications of trigonometry, where heights and distances are measured, but also lays the foundation for the vast subject of trigonometry used throughout science and engineering. Mastery of the ratios, identities, and standard values ensures a strong start to one of the most scoring units in the board examination.