Trigonometric Functions: Class 11 Maths · Chapter 3

3.1 Introduction

At a Dussehra mela, keep your eye on one seat of the giant wheel. The wheel turns at a steady rate and the seat rises, falls and rises again, repeating the same pattern of heights on every turn. A right triangle cannot describe this. Once the seat passes the top, it has turned through more than ${90}^{\circ },$ and after a few rounds through more than ${1000}^{\circ }\mathrm{.}$

Anything that goes round, or swings to and fro, makes a pattern like this. The 50-hertz alternating current in Indian homes, the tide rising and falling at Mumbai, a vibrating sitar string and a satellite circling the Earth are all described with the functions of this chapter. Their parent subject, trigonometry (त्रिकोणमिति), still carries its older job in its name: the Greek words behind it mean triangle and measure.

Your Class 10 tools cannot follow the seat for long. There, $\sin \theta$ was opposite over hypotenuse in a right triangle, which only makes sense while $\theta$ is acute. We keep everything that worked, the values at ${30}^{\circ },$ ${45}^{\circ }$ and ${60}^{\circ }$ and the identity ${\sin }^2\theta +{\cos }^2\theta =1,$ and remove the limit on the angle. The plan: measure angles as amounts of turning (in degrees and in a new unit, the radian), define sine and cosine for every real number with a circle, draw their graphs, and find the sine and cosine of a sum of two angles.

3.2 Angles

Picture a door swinging on its hinge. In this chapter an angle is an amount of turning like that: a ray starts in one position, called its initial side, and turns about its endpoint, the vertex, until it reaches its terminal side. A sign records the direction, positive for anticlockwise turning and negative for clockwise. Nothing stops the ray from going round more than once, so an angle may be larger than a full turn (Fig. 3.2).

Turning can be counted in whole turns, or revolutions; motors and fans are rated this way, in revolutions per minute, and a ceiling fan on full speed manages a few hundred. Whole turns are too coarse for most geometry, so two finer units are used instead: the degree and the radian.

3.2.1 Degree measure

One degree (अंश), written $1^{\circ },$ is $\frac{1}{360}$ of a full turn. Finer still are the minute, a sixtieth of a degree, and the second, a sixtieth of a minute:

A full turn is therefore ${360}^{\circ },$ half a turn ${180}^{\circ }$ and a quarter turn ${90}^{\circ }\mathrm{.}$ An angle of ${450}^{\circ }$ is a full turn followed by a quarter turn, while $-{135}^{\circ }$ is three-eighths of a turn clockwise. Minutes and seconds still appear in astronomy and map-making; the Earth's axis, for example, is tilted at about ${23}^{\circ }{26}^{\mathrm{\prime }}$ to the perpendicular of its orbit.

3.2.2 Radian measure

The number 360 is a human choice with no special meaning for a circle. A unit taken from the circle itself is more useful. Lay a string exactly as long as the radius along the rim (Fig. 3.3). The angle it covers at the centre is the same for every circle, large or small, and it is called one radian.

The circumference is $2\pi r,$ so exactly $2\pi$ radius-lengths fit round the rim, and one complete turn measures $2\pi$ radians. The angle at the centre grows in step with the arc it cuts off, so an arc of length $l$, which is $l\mathrm{/}r$ radius-lengths long, subtends $l\mathrm{/}r$ radians. This is the arc-length formula:

3.2.3 Relation between radian and real numbers

Radians let us treat angles as plain real numbers. Draw the unit circle with centre $O$ and mark $A=(1,0)\mathrm{.}$ Stand a number line at $A$ so that it touches the circle, with $0$ at $A$ and the positive numbers going up (Fig. 3.4). Now wrap the positive half anticlockwise round the circle and the negative half clockwise, like thread round a spool.

Each real number $t$ lands on one point of the circle, and the arc from $A$ to that point has length $\mathrm{\mid }t\mathrm{\mid }$. So $t$ marks an angle of $t$ radians: positive numbers give anticlockwise angles and negative numbers clockwise ones. Different numbers can land on the same point, such as $t$ and $t+2\pi ,$ but every real number is the radian measure of exactly one angle and every angle has a radian measure. That is why we can feed any real number into sine and cosine.

3.2.4 Relation between degree and radian

A full turn is ${360}^{\circ }$ and also $2\pi$ radians. Halving both gives the one fact you need for every conversion:

Dividing, $1$ rad $={\left(\frac{180}{\pi }\right)}^{\circ }\approx {57.2958}^{\circ },$ which is about ${57}^{\circ }{17}^{\mathrm{\prime }}{45}^{\mathrm{\prime }\mathrm{\prime }},$ and $1^{\circ }=\frac{\pi }{180}$ rad, about $0.01745$ rad. With the rough value $\pi \approx \frac{22}{7}$ the first figure comes out as about ${57}^{\circ }{16}^{\mathrm{\prime }},$ which is close enough for most problems. To convert, multiply by the right fraction:

3.3 Trigonometric Functions

Place a right triangle so that its acute angle $x$ sits at the centre of a circle of radius 1 and its hypotenuse is a radius. The hypotenuse is 1, so $\cos x$ (adjacent over hypotenuse) equals the horizontal coordinate of the corner on the circle, and $\sin x$ equals its vertical coordinate. That observation tells us how to define sine and cosine when there is no triangle at all: read them off the circle.

Let the unit circle have its centre $O$ at the origin, and let $A=(1,0)\mathrm{.}$ For any real number $x$, travel round the circle from $A$ through an arc of length $x$, anticlockwise if $x>0$ and clockwise if $x<0,$ and call the point you reach $P(a,b)\mathrm{.}$ The angle $AOP$ is then $x$ radians (Fig. 3.5).

Triangle $OMP$ has a right angle at $M$, with $OM=\mathrm{\mid }a\mathrm{\mid },$ $MP=\mathrm{\mid }b\mathrm{\mid }$ and $OP=1.$ Pythagoras gives $a^2+b^2=1$ for every point of the circle, in every quadrant, because squaring removes the signs. In the new notation:

Here ${\cos }^{2}x$ means $(\cos x{)}^2\mathrm{.}$ The identity holds for every real $x$, not just for acute angles.

The last step of Fig. 3.5 marks the four points where the circle meets the axes. They are reached by whole-number multiples of $\frac{\pi }{2},$ which are therefore called quadrantal angles, and their sines and cosines can simply be read off. The pair $(\cos x,\sin x)$ is $(1,0)$ at $x=0,$ $(0,1)$ at $x=\frac{\pi }{2},$ $(-1,0)$ at $x=\pi$ and $(0,-1)$ at $x=\frac{3\pi }{2}$; at $x=2\pi$ we are back at $(1,0)\mathrm{.}$

An arc of length $x+2\pi$ is an arc of length $x$ followed by one extra lap, and a lap ends where it began. The same holds for any whole number of laps, forwards or backwards, so

The sine is the height of $P$, so it is zero exactly when $P$ is at $A$ or $C$, that is, when $x$ is a multiple of $\pi$. The cosine vanishes exactly when $P$ is at $B$ or $D$, that is, when $x$ is $\frac{\pi }{2}$ times an odd integer:

The other four functions are built from sine and cosine. Each one is defined wherever its denominator is not zero:

Two more Pythagorean identities follow at once. Divide (3.4) by ${\cos }^{2}x,$ which is allowed whenever $\cos x\mathrm{\ne }0$; then divide it instead by ${\sin }^{2}x,$ allowed whenever $\sin x\mathrm{\ne }0$:

For $0<x<\frac{\pi }{2}$ the triangle $OMP$ is exactly the right triangle of Class 10, so the new definitions agree with the old ratios and the familiar values carry over (Table 3.2). A handy way to remember the sine column: it reads $\frac{\sqrt{0}}{2},\frac{\sqrt{1}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2},\frac{\sqrt{4}}{2},$ and the cosine column is the same list in reverse.

For the reciprocal functions, flip the entries: for instance $\sec \frac{\pi }{3}=2$ and $\cot \frac{\pi }{6}=\sqrt{3},$ while $\mathrm{cosec}0$ does not exist.

3.3.1 Sign of trigonometric functions

Compare the point $P(a,b)$ at arc length $x$ with the point $Q$ at arc length $-x$. The two arcs are equally long but run in opposite directions from $A$, so $Q$ is the mirror image of $P$ in the $x$-axis: $Q=(a,-b)$ (Fig. 3.6). This works in every quadrant, so for all real $x$:

A function with $f(-x)=f(x)$ for every $x$ is an even function; one with $f(-x)=-f(x)$ is an odd function. So cosine is even and sine is odd. Dividing the two rules shows that $\tan (-x)=-\tan x,$ so tangent is odd as well.

Because $P$ never leaves the unit circle, neither coordinate can exceed 1 in size: $\mathrm{\mid }\sin x\mathrm{\mid }\le 1$ and $\mathrm{\mid }\cos x\mathrm{\mid }\le 1$ for every $x$. The signs of the two coordinates depend only on the quadrant that $P$ is in, and the other four functions, being built from sine and cosine, inherit their signs. Fig. 3.7 walks round the four quadrants.

The same reasoning works on a circle of any radius $r$. If $P(a,b)$ lies on that circle and $OP$ makes an angle $x$ with the positive $x$-axis, then $\cos x=\frac{a}{r}$ and $\sin x=\frac{b}{r},$ because triangle $OMP$ is the unit-circle triangle scaled up by $r$. The simulation uses $r=5,$ where grid points such as $(3,4)$ and $(-4,3)$ give exact fractions.

3.3.2 Domain and range of trigonometric functions

Sine and cosine are defined for every real number, since every $x$ gives a point on the circle, and their values are coordinates of that point, so both have range $[-1,1]\mathrm{.}$ The other four functions lose the points where their denominators vanish. Because the reciprocal of a number in $[-1,1]$ lies outside $(-1,1),$ secant and cosecant never take values strictly between $-1$ and $1$. Table 3.4 collects the results; $n$ stands for any integer.

As $x$ runs from $0$ to $2\pi$, the point $P$ goes once round the circle, and you can read off how each function changes. The height $\sin x$ climbs from $0$ to $1$ in quadrant I, falls back to $0$ in II, drops to $-1$ in III and returns to $0$ in IV. The cosine starts at $1$, falls steadily to $-1$ at $x=\pi$ and then climbs back to $1$. The tangent increases in every quadrant: from $0$ it grows without bound as $x$ approaches $\frac{\pi }{2}$,11Writing $\tan x\to \mathrm{\infty }$ is shorthand for this unbounded growth. $\mathrm{\infty }$ is not a number, and $\tan x$ never equals it. then returns from very large negative values just after $\frac{\pi }{2}$ and rises again.

A function $f$ is periodic if there is a number $p>0$ with $f(x+p)=f(x)$ for every $x$ in its domain; the smallest such $p$ is its period. Sine and cosine have period $2\pi$, and so do their reciprocals. Tangent and cotangent repeat sooner, after $\pi$: half a turn sends $P(a,b)$ to $(-a,-b),$ which leaves the ratio $\frac{b}{a}$ unchanged. We prove $\tan (\pi +x)=\tan x$ in Section 3.4.

Plotting the values gives the graphs in Fig. 3.9, Fig. 3.10 and Fig. 3.11. Where a function is undefined, its graph runs alongside a vertical line, getting closer and closer without ever touching it. Such a line is called an asymptote.