Chapter 3 ยท Introduction

Chapter 3 โ€” Trigonometric Functions

Turn the ratios of a right triangle into functions of any real number โ€” measure angles in radians, wind a point round the unit circle, and discover the identities that tie sine and cosine together.

A Ferris wheel turns, and the height of your seat rises and falls, rises and falls, every few minutes. The tide at Mumbai's harbour comes in and goes out roughly twice a day; the voltage in the socket on your wall swings from positive to negative fifty times every second; the length of the day in Delhi grows and shrinks over a year. All of these repeat. In Class 10 you met sine, cosine and tangent as ratios of sides of a right-angled triangle, which only makes sense for angles between $0^\circ$ and $90^\circ$. To describe a wheel that keeps on turning, we need sine and cosine of $200^\circ$, of $-45^\circ$, of $1000^\circ$ โ€” of any angle at all.

This chapter makes that leap. First we measure angles in a new, natural unit, the radian, in which an angle is simply "how many radius-lengths of arc it cuts off". Then we place a point on a circle of radius $1$, let it travel a distance $x$ round the circle, and define $\cos x$ and $\sin x$ as the coordinates where it lands. These are the trigonometric functions of a real number: they agree with your Class 10 ratios for acute angles, but they are defined for every real $x$, they repeat every $2\pi$, and their graphs are the waves you see on an oscilloscope. Finally we prove the sum and difference formulae โ€” the single most useful set of identities in school trigonometry โ€” and derive from them the double-angle, triple-angle and productโ€“sum formulae that you will use again and again in calculus, physics and JEE problems.

Measure: degrees, radians, $s=r\theta$โ†’Define: $\cos x,\sin x$ on the unit circleโ†’Combine: $\sin(x\pm y)$, $\cos(x\pm y)$โ†’Multiply: $\sin2x$, $\cos3x$, product โ†” sum
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • describe positive and negative angles, and convert between degree and radian measure using $\pi$ radians $=180^\circ$;
  • use $s=r\theta$ to find arc lengths, radii and angles in clocks, wheels and pendulums;
  • define the six trigonometric functions of a real number using the unit circle, and find their signs in each quadrant;
  • state the domain, range and period of each trigonometric function, sketch its graph, and evaluate it at any angle by reducing to an acute angle;
  • prove and apply the sum and difference formulae for sine, cosine and tangent;
  • use the formulae for $\sin2x$, $\cos2x$, $\tan2x$, $\sin3x$, $\cos3x$, $\tan3x$ and the product โ†” sum formulae to evaluate expressions and prove identities.
Teaser โ€” Unrolling the Unit Circle

A point $P$ walks anticlockwise round a circle of radius $1$. When it has walked a distance $x$ along the circle, its height is $\sin x$ and its horizontal position is $\cos x$. On the right, the distance $x$ is laid out along the horizontal axis and the chosen coordinate is plotted above it โ€” the circle "unrolls" into a wave that repeats every $2\pi\approx6.28$.

arc length $x$โ€”
in degreesโ€”
$\cos x$โ€”
$\sin x$โ€”

Sections in this chapter

3.1

Angles: Degree and Radian Measure

Positive and negative angles, degrees, minutes and seconds, the radian, conversion, and arc length.

$\pi\ \text{rad}=180^\circ,\ \ s=r\theta$
Open section โ†’3.2

Trigonometric Functions of a Real Number

Sine and cosine on the unit circle, the six functions, signs by quadrant, domains, ranges, periods and graphs.

$\sin^2x+\cos^2x=1$
Open section โ†’3.3

Sum and Difference Formulae

A proof of $\cos(x+y)$, the formulae for $\sin(x\pm y)$, $\tan(x\pm y)$, and angles such as $\pi\pm x$.

$\sin(x+y)=\sin x\cos y+\cos x\sin y$
Open section โ†’3.4

Multiple-Angle, Product and Sum Formulae

Double and triple angles, products turned into sums and sums into products, and proving identities.

$\cos2x=1-2\sin^2x$
Open section โ†’
โœ… Before you start
From Class 10 Mathematics you need the trigonometric ratios of an acute angle in a right-angled triangle, their values at $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$, and the identity $\sin^2A+\cos^2A=1$ (Class 10, Section 8.1 onwards), together with the circumference $2\pi r$ of a circle and the distance formula of coordinate geometry. From this course you need the idea of a real function with a domain and a range (Section 2.3) and the algebra of functions (Section 2.4) โ€” in this chapter $\sin$, $\cos$ and $\tan$ become real functions in exactly that sense.
โ† 2.4 Algebra of Real Functions3.1 Angles: Degree and Radian Measure โ†’