3.2 Trigonometric Functions of a Real Number
Wind a point round a circle of radius $1$: its coordinates after travelling a distance $x$ are $\cos x$ and $\sin x$. This one picture defines all six trigonometric functions for every real number, and explains their signs, ranges, periods and graphs.
In Class 10, $\sin A$ was "opposite over hypotenuse" in a right-angled triangle โ a definition that only works for $0^\circ\lt A\lt90^\circ$, because a right triangle cannot have an obtuse angle, let alone an angle of $-45^\circ$ or $750^\circ$. Yet the height of a seat on a Ferris wheel, the voltage of the mains supply and the position of a piston all depend on angles that keep growing. We need a definition of sine and cosine that works for every real number and agrees with the old one for acute angles.
The idea is beautifully simple: draw a circle of radius $1$ centred at the origin, start at the point $(1,0)$, and walk a distance $x$ round the circle. Wherever you stop, your $x$-coordinate is $\cos x$ and your $y$-coordinate is $\sin x$. In this section we make that precise, define the other four functions from sine and cosine, and work out their signs in each quadrant, their domains and ranges, and their graphs โ waves that repeat forever.
Sine and Cosine on the Unit Circle
The unit circle is the circle $a^2+b^2=1$ of radius $1$ centred at the origin $O$. Let $x$ be any real number. Starting at $A(1,0)$, move along the unit circle a distance $|x|$ โ anticlockwise if $x\gt0$, clockwise if $x\lt0$ โ and let $P(a,b)$ be the point where you stop. Then
$$\cos x=a,\qquad \sin x=b .$$Equivalently, by Definition 3.3, $\angle AOP$ is an angle of $x$ radians in standard position, and $P$ is the point where its terminal side meets the unit circle. So $\cos x$ and $\sin x$ are the coordinates of $P$: $\ P=(\cos x,\sin x)$.
Agreement with Class 10. If $0\lt x\lt\tfrac{\pi}{2}$, drop the perpendicular $PM$ to the $x$-axis. Triangle $OMP$ has a right angle at $M$, hypotenuse $OP=1$, and angle $x$ at $O$. Its Class 10 ratios are $\tfrac{OM}{OP}=a$ and $\tfrac{MP}{OP}=b$ โ exactly the new $\cos x$ and $\sin x$. So nothing you learnt is lost; the definition has simply been extended.
For every real number $x$ and every integer $n$:
- $\sin^2x+\cos^2x=1$;
- $-1\le\sin x\le1$ and $-1\le\cos x\le1$;
- $\sin(x+2n\pi)=\sin x$ and $\cos(x+2n\pi)=\cos x$;
- $\sin(-x)=-\sin x$ and $\cos(-x)=\cos x$;
- $\sin x=0$ exactly when $x=n\pi$, and $\cos x=0$ exactly when $x=(2n+1)\tfrac{\pi}{2}$.
Proof. (1) $P(\cos x,\sin x)$ lies on $a^2+b^2=1$. (2) Then $\sin^2x=1-\cos^2x\le1$, so $|\sin x|\le1$; similarly for $\cos x$. (3) Adding $2n\pi$ adds $n$ complete laps of the circle (circumference $2\pi$), so we stop at the same $P$. (4) Walking the distance $|x|$ clockwise instead of anticlockwise lands at the mirror image of $P$ in the $x$-axis, $(a,-b)$. (5) $\sin x=0$ when $P$ is on the $x$-axis, i.e. at $(1,0)$ or $(-1,0)$, which we reach after $0,\pi,2\pi,\dots$ or $-\pi,-2\pi,\dots$ โ the multiples of $\pi$. $\cos x=0$ when $P$ is $(0,\pm1)$, reached at the odd multiples of $\tfrac{\pi}{2}$. $\blacksquare$
We write $\sin^2x$ for $(\sin x)^2$ โ not $\sin(x^2)$. Property (3) says that sine and cosine repeat every $2\pi$; property (4) says cosine is an even function and sine is an odd function. The quadrantal points give these values:
| $x$ | $0$ | $\tfrac{\pi}{2}$ | $\pi$ | $\tfrac{3\pi}{2}$ | $2\pi$ |
|---|---|---|---|---|---|
| $P$ | $(1,0)$ | $(0,1)$ | $(-1,0)$ | $(0,-1)$ | $(1,0)$ |
| $\cos x$ | $1$ | $0$ | $-1$ | $0$ | $1$ |
| $\sin x$ | $0$ | $1$ | $0$ | $-1$ | $0$ |
The Six Functions and Their Signs
For real $x$ and $n\in\mathbb{Z}$, wherever the denominator is not zero:
$$\tan x=\frac{\sin x}{\cos x}\ \ \left(x\ne(2n+1)\tfrac{\pi}{2}\right),\qquad \cot x=\frac{\cos x}{\sin x}\ \ (x\ne n\pi),$$ $$\sec x=\frac{1}{\cos x}\ \ \left(x\ne(2n+1)\tfrac{\pi}{2}\right),\qquad \operatorname{cosec}x=\frac{1}{\sin x}\ \ (x\ne n\pi).$$These are the tangent, cotangent, secant and cosecant functions. Together with sine and cosine they are the six trigonometric functions.
For every real $x$ at which both sides are defined,
$$1+\tan^2x=\sec^2x\quad\left(x\ne(2n+1)\tfrac{\pi}{2}\right),\qquad 1+\cot^2x=\operatorname{cosec}^2x\quad(x\ne n\pi).$$Proof. Divide $\sin^2x+\cos^2x=1$ by $\cos^2x$ (allowed, since $\cos x\ne0$): $\tan^2x+1=\sec^2x$. Divide instead by $\sin^2x\ne0$: $1+\cot^2x=\operatorname{cosec}^2x$. $\blacksquare$
The sign of each function depends only on the quadrant containing $P$. In Quadrant I both coordinates are positive, so all six functions are positive. In Quadrant II, $a\lt0$ and $b\gt0$: only $\sin$ (and its reciprocal $\operatorname{cosec}$) is positive. In Quadrant III both are negative, so their ratio $\tan$ (and $\cot$) is positive. In Quadrant IV only $\cos$ (and $\sec$) is positive.
Signs. Going anticlockwise from Quadrant I: All, Sine, Tangent, Cosine are positive (memory aid: "Add Sugar To Coffee"). Reciprocals share signs: $\operatorname{cosec}$ with $\sin$, $\sec$ with $\cos$, $\cot$ with $\tan$.
- From the given value, use $\sin^2x+\cos^2x=1$, $1+\tan^2x=\sec^2x$ or $1+\cot^2x=\operatorname{cosec}^2x$ to find the size of a second function.
- Choose its sign from the quadrant (ASTC) โ this is the step students forget.
- Get the remaining functions as quotients and reciprocals, and check one of them with an identity.
The lab below also shows the reference angle โ the acute angle between the terminal side and the $x$-axis. By the symmetry of the circle the values at $\theta$ have the same size as at the reference angle; Section 3.3 proves this with the allied-angle formulae.
Domain, Range, Period and Graphs
A function $f$ is periodic if there is a positive number $T$ such that $f(x+T)=f(x)$ for every $x$ in its domain. The smallest such $T$ is the period of $f$. The graph of a periodic function is one piece, of width $T$, repeated forever to the left and to the right.
$\sin$, $\cos$, $\sec$ and $\operatorname{cosec}$ have period $2\pi$; $\tan$ and $\cot$ have period $\pi$:
$$\tan(x+\pi)=\tan x,\qquad \cot(x+\pi)=\cot x .$$Proof. By Theorem 3.3, $2\pi$ works for sine and cosine, hence for their reciprocals. No smaller positive $T$ works for sine: $\sin\left(\tfrac{\pi}{2}+T\right)=\sin\tfrac{\pi}{2}=1$ would need the point for $\tfrac{\pi}{2}+T$ to be $(0,1)$ again, which first happens after a full lap. For tangent: walking an extra distance $\pi$ (half a lap) takes $P(a,b)$ to the diametrically opposite point $(-a,-b)$, so $\tan(x+\pi)=\dfrac{-b}{-a}=\dfrac ba=\tan x$. Since $\tan x=0$ only at $x=n\pi$, no positive $T\lt\pi$ can work. $\blacksquare$
| function | domain | range | period |
|---|---|---|---|
| $\sin x$ | $\mathbb{R}$ | $[-1,1]$ | $2\pi$ |
| $\cos x$ | $\mathbb{R}$ | $[-1,1]$ | $2\pi$ |
| $\tan x$ | $\mathbb{R}-\left\{(2n+1)\tfrac{\pi}{2} : n\in\mathbb{Z}\right\}$ | $\mathbb{R}$ | $\pi$ |
| $\cot x$ | $\mathbb{R}-\{n\pi : n\in\mathbb{Z}\}$ | $\mathbb{R}$ | $\pi$ |
| $\sec x$ | $\mathbb{R}-\left\{(2n+1)\tfrac{\pi}{2} : n\in\mathbb{Z}\right\}$ | $(-\infty,-1]\cup[1,\infty)$ | $2\pi$ |
| $\operatorname{cosec}x$ | $\mathbb{R}-\{n\pi : n\in\mathbb{Z}\}$ | $(-\infty,-1]\cup[1,\infty)$ | $2\pi$ |
The ranges of $\sec$ and $\operatorname{cosec}$ follow from $|\cos x|\le1$: the reciprocal of a number between $-1$ and $1$ (other than $0$) has size at least $1$. As $P$ moves round the circle, the values change like this:
| Quadrant I | Quadrant II | Quadrant III | Quadrant IV | |
|---|---|---|---|---|
| $\sin x$ | increases $0\to1$ | decreases $1\to0$ | decreases $0\to-1$ | increases $-1\to0$ |
| $\cos x$ | decreases $1\to0$ | decreases $0\to-1$ | increases $-1\to0$ | increases $0\to1$ |
| $\tan x$ | increases $0\to\infty$ | increases $-\infty\to0$ | increases $0\to\infty$ | increases $-\infty\to0$ |
Where $\cos x=0$, $\tan x$ and $\sec x$ are undefined and their graphs have vertical asymptotes โ lines the graph approaches but never meets. Likewise $\cot x$ and $\operatorname{cosec}x$ have asymptotes where $\sin x=0$.
(b) For which $x$ in $[0,2\pi]$ is $\tan x$ not defined?
(c) Show that $g(x)=\sin2x$ has period $\pi$.
1. $\sin^2x$ is not $\sin x^2$. $\sin^2x=(\sin x)^2$; $\sin x^2=\sin(x^2)$. And $\sin^{-1}x$ does not mean $\tfrac{1}{\sin x}$ โ write $\operatorname{cosec}x$ or $(\sin x)^{-1}$.
2. Forgetting the sign. $\sin^2x=\tfrac{16}{25}$ gives $\sin x=\pm\tfrac45$; only the quadrant decides which.
3. "$\sin x$" means $x$ in radians. $\sin2$ is the sine of $2$ radians ($\approx114.6^\circ$), a positive number about $0.909$ โ not $\sin2^\circ\approx0.035$.
4. Impossible values. $\sin x=1.2$ and $\sec x=0.5$ have no solutions; $\tan x$ and $\cot x$, on the other hand, take every real value.
5. The period of $\tan$ is $\pi$, not $2\pi$. $2\pi$ also "works", but the period is the smallest positive value that works.
Exercises
- 1 $\sin\tfrac{3\pi}{2}+\cos\pi$
- 2 $\cos 7\pi$
- 3 $\sin\left(-\tfrac{7\pi}{2}\right)$
- 4 $\sin405^\circ$
- 5 $\cos(-1050^\circ)$
- 6 $\tan\tfrac{13\pi}{4}$
- 7 $\sec\tfrac{25\pi}{3}$
- 8 $\operatorname{cosec}(-1110^\circ)$
- 9 $\cos(-9\pi)$
- 10 $\cot\tfrac{17\pi}{4}+\tan\left(-\tfrac{11\pi}{4}\right)$
- 1 Find the sign of $\sin200^\circ\cdot\cos300^\circ$.
- 2 Find the sign of $\tan3$.
- 3 Find the sign of $\cos5$.
- 4 $\sin x=\tfrac35$, $x$ in Quadrant II. Find the other five values.
- 5 $\tan x=-\tfrac{5}{12}$, $x$ in Quadrant IV. Find the other five values.
- 6 $\sec x=\tfrac{13}{5}$, $x$ in Quadrant IV. Find the other five values.
- 7 $\cot x=\tfrac34$, $x$ in Quadrant III. Find the other five values.
- 8 $\sin x=-\tfrac12$ and $\tan x\gt0$. In which quadrant is $x$, and what is $\cos x$?
- 1 Prove that $(1-\sin^2x)\sec^2x=1$, for $x\ne(2n+1)\tfrac{\pi}{2}$.
- 2 Prove that $\sin^4x-\cos^4x=\sin^2x-\cos^2x$.
- 3 Prove that $(\sec x-\tan x)(\sec x+\tan x)=1$, where defined.
- 4 Prove that $\tan x+\cot x=\sec x\operatorname{cosec}x$, where defined.
- 5 Find the range of $f(x)=5-2\sin x$.
- 6 Find the greatest and least values of $3-4\cos x$, and where in $[0,2\pi]$ they occur.
- 7 Which of these equations have solutions? (a) $\sin x=\tfrac43$ (b) $\sec x=\tfrac12$ (c) $\tan x=250$ (d) $\operatorname{cosec}x=-1$.
- 8 Find all $x$ in $[0,2\pi]$ with $\cos x=0$, and all $x$ in $[0,2\pi]$ with $\sin x=0$.
- 1 State the period of $\cot x$, and the points of $(-\pi,\pi)$ where $\tan x$ is not defined.
- 2 On which interval within $[0,2\pi]$ does $\cos x$ decrease, and on which does it increase?
- 3 Show that $\cos3x$ is periodic with period $\tfrac{2\pi}{3}$ (you may assume no smaller period works).
- 4 Giant wheel. The height of a cabin above the ground is $h(t)=20-18\cos\tfrac{\pi t}{5}$ metres, $t$ minutes after it passes the lowest point. Find $h(0)$, $h(5)$, the greatest and least heights, and the time for one revolution.
- 5 Tides. The depth of water at a harbour entrance is $d(t)=6+2\sin\tfrac{\pi t}{6}$ metres, $t$ hours after midnight. Find the greatest and least depths, the first time each occurs, and the period.
- 6 Sketch $y=\sin x$ and $y=\cos x$ on the same axes for $0\le x\le2\pi$, and find the values of $x$ where the graphs cross.
How this is examined: the Board's favourite is "given one ratio and the quadrant, find the other five" (3 marks), plus values at large or negative angles (2 marks) and periodic models. JEE uses this section inside bigger problems: signs of $\sin2$ or $\tan4$, the range of an expression, or a value forced by $|\sin x|\le1$.
E1. Board ยท 1 mark If $\sin x=-\tfrac45$ and $x$ lies in the third quadrant, then $\tan x=$ (A) $\tfrac43$ (B) $-\tfrac43$ (C) $\tfrac34$ (D) $-\tfrac34$
E2. Board ยท 2 marks Find the values of (i) $\sin(-1125^\circ)$ (ii) $\cot\tfrac{49\pi}{4}$.
E3. Board ยท 3 marks If $\sec x=-\tfrac{13}{12}$ and $x$ lies in the third quadrant, find the values of the other five trigonometric functions.
E4. Board ยท Case study A cabin on a giant wheel is $h(t)=22-20\cos\tfrac{\pi t}{10}$ metres above the ground, $t$ minutes after it passes the lowest point. (i) Find the least and greatest heights of the cabin. (2 marks) (ii) How long does one revolution take? (1 mark) (iii) Find the height after $5$ minutes. (1 mark)
E5. JEE Main If $\sin x+\operatorname{cosec}x=2$, then $\sin^{2026}x+\operatorname{cosec}^{2026}x$ equals (A) $2$ (B) $2^{2026}$ (C) $1$ (D) $0$
E6. JEE Advanced One or more options correct. Which of the following numbers are positive? (A) $\sin2$ (B) $\cos2$ (C) $\tan4$ (D) $\sec5$
Research question. How well does a sine curve describe the length of the day in your town over a year?
Why it matters. Day length controls crop cycles, solar-panel output and energy demand. Climate scientists and power-grid planners model it โ and many other seasonal quantities โ with functions of the form $A+B\sin\tfrac{2\pi(t-c)}{365}$, because the Earth's orbit repeats every year.
What to do.
- From a weather app or almanac website, record the sunrise and sunset times in your town on the 1st and 15th of every month (24 values), and compute the day length in hours.
- Plot day length against day number $t$ ($1$ to $365$) in a spreadsheet.
- Estimate $A$ (the average day length), $B$ (half the difference between the longest and shortest days) and $c$ (the day on which the length is average and increasing, near 21 March).
- Plot $L(t)=A+B\sin\tfrac{2\pi(t-c)}{365}$ on the same axes and list the largest gap between model and data.
- Repeat with data for a city much farther from the equator (e.g. Oslo or Helsinki) and compare the values of $B$.
Tools: Weather or almanac website, spreadsheet with a chart
Hand in: Your data table, the graph with the fitted curve, the values of $A$, $B$, $c$, and three sentences on what the period and the amplitude mean physically.
Going further. Why is the period $365$ days and not $\tfrac{365}{2}$? What would the graph look like at the equator?
๐งช Self-check quiz
Q1. $\cos(-\pi)$ equals
Q2. In which quadrant is $\sin x$ negative and $\cos x$ positive?
Q3. The period of $\tan x$ is
Q4. The range of $\sec x$ is
Q5. If $\tan x=\tfrac34$ and $x$ lies in the third quadrant, then $\cos x=$