MATH 1.6: Trigonometric functions	PPLATO @
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1.1 Module introduction

Trigonometric functions have a wide range of application in physics; examples include the addition and resolution of vectors (such as forces), the description of simple harmonic motion and the formulation of quantum theories of the atom. Trigonometric functions are also important for solving certain differential equations, a topic which is considered in some detail elsewhere in FLAP.

In Section 2 of this module we begin by looking at the measurement of angles in degrees and in radians. We then discuss some basic ideas about triangles, including Pythagoras’s theorem, and we use right–angled triangles to introduce the trigonometric ratios (sinθ, cosθ and tanθ) and the reciprocal trigonometric ratios (secθ, cosecθ and cotθ). In Section 3 we extend this discussion to include the trigonometric functions (sin(θ), cos(θ) and tan(θ)) and the reciprocal trigonometric functions (cosec(θ), sec(θ) and cot(θ).) These periodic functions generalize the corresponding ratios since the argument θ may take on values that are outside the range 0 to π/2. Subsection 3.2 discusses the related inverse trigonometric functions (arcsin(x), arccos(x) and arctan(x)), paying particular attention to the conditions needed to ensure they are defined. We end, in Section 4, by showing how the sides and angles of any triangle are related by the sine rule and the cosine rule and by listing some useful identities involving trigonometric functions.

Study comment Having read the introduction you may feel that you are already familiar with the material covered by this module and that you do not need to study it. If so, try the following Fast track questions. If not, proceed directly to the Subsection 1.3Ready to study? Subsection.

1.2 Fast track questions

Study comment Can you answer the following Fast track questions? If you answer the questions successfully you need only glance through the module before looking at the Subsection 5.1Module summary and the Subsection 5.2Achievements. If you are sure that you can meet each of these achievements, try the Subsection 5.3Exit test. If you have difficulty with only one or two of the questions you should follow the guidance given in the answers and read the relevant parts of the module. However, if you have difficulty with more than two of the Exit questions you are strongly advised to study the whole module.

1.3 Ready to study?

Study comment In order to study this module you will need to understand the following terms: constant, decimal places, function, power_mathematicalpower, reciprocal and variable. An understanding of the terms codomain, domain and inverse function would also be useful but is not essential. You will need to be able to solve a pair of simultaneous_linear_equationssimultaneous equations, manipulate arithmetic and algebraic expressions – including those involving square_of_a_numbersquares, square roots, brackets and ratios – and to plot and interpret graphs. If you are uncertain about any of these terms, you should consult the Glossary, which will also indicate where in FLAP they are developed. The following questions will help you to decide whether you need to review some of these topics before embarking on this module.

2 Triangles and trigonometric ratios

2.1 Angular measure: degrees and radians

When two straight lines pass through a common point, the angle between the lines provides a measure of the inclination of one line with respect to the other or, equivalently, of the amount one line must be rotated about the common point in order to make it coincide with the other line. The two units commonly used to measure angles are degrees and radians (discussed below) and we will use both throughout this module. Greek letters, α (alpha), β (beta), γ (gamma), ... θ (theta), ϕ (phi) ... are often used to represent the values of angles, but this is not invariably the case.

A degree is defined as the unit of angular measure corresponding to 1/360th of a circle and is written as 1°. In other words, a rotation through 360° is a complete revolution, and an object rotated through 360° about a fixed point is returned to its original position. Fractions of an angle measured in degrees are often expressed as decimals, as in 97.8°, but it is also possible to use subsidiary units usually referred to as minutes and seconds for subdivisions of a degree, with the definitions that sixty minutes are equivalent to one degree and sixty seconds are equivalent to one minute. To distinguish them from units of time, these angular units are called the minute of arc and second of arc, abbreviated to arcmin and arcsec, respectively. i The symbols ′ and ′′ are often used for arcmin and arcsec, respectively. For example, 12′ means 12 arcmin, and 35′′ means 35 arcsec.

The angles 180° and 90° correspond to a rotation through half and one–quarter of a circle, respectively. An angle of 90° is known as a right angle. A line at 90° to a given line (or surface) is said to be perpendicular or normal to the original line (or surface).

It is conventional in mathematics and physics to refer to rotations in an anti–clockwise direction as positive rotations. So, a positive rotation through an angle θ would correspond to the anticlockwise movement shown in Figure 1, while a negative rotation of similar size would correspond to a clockwise movement. The negative rotation may be described as a rotation through an angle −θ.

An object which is rotated through an angle of 0° or 360° or 720° appears to remain unchanged and, in this sense, these rotations are equivalent. In the same sense rotations of 10°, 370°, 730°, −350° and so on are equivalent, since each can be obtained from the others by the addition of a multiple of 360°. When considering the orientational effect of a rotation through an angle θ it is only necessary to consider values of θ which lie in the range 0° ≤ θ < 360° i since the orientational effect of every rotation is equivalent to a rotation lying in this range.

For example, a rotation through −1072° is equivalent to one through −1072° + 3 × 360° = 8°. Notice that the range of non–equivalent rotations, 0° ≤ θ < 360°, does not include 360°. This is because a rotation through 360° is equivalent to one through 0°, which is included.

Of course, angles that differ by a multiple of 360° are not equivalent in every way. For example, a wheel that rotates through 36 000° and thus completes 100 revolutions would have done something physically different from an identical wheel that only rotated through an angle of 360°, even if their final orientations were the same.

Despite the widespread use of degrees, a more natural (and important) unit of angular measure is the radian. As Figure 2 indicates, the radian may be defined as the angle subtended at the centre of a circle by an arc of the circle which has an arc length i equal to the radius of the circle. As will be shown below, it follows from this definition that 1 radian (often abbreviated as 1 rad or sometimes 1^c) is equal to 57.30°, to two decimal places.

Radians are such a natural and widely used unit of angular measure that whenever you see an angular quantity quoted without any indication of the associated unit of measurement, you should assume that the missing unit is the radian.

This is a sensible definition of an angle since it is independent of the scale of Figure 3. For a given value of ϕ, a larger value of r would result in a larger value of s but the ratio s/r would be unchanged.

Table 1 Equivalent angles in degrees and radians.

Angle measured in degrees	Angle measured in radians
360	2π
180	π
90	π/2
60	π/3
45	π/4
30	π/6

To determine the fixed relationship between radians and degrees it is probably easiest to consider the angle subtended at the centre of a circle by the complete circumference of that circle. A circle of radius r has a circumference of arc length 2πr, where π represents the mathematical constant pi, an irrational number i the value of which is 3.1416 to four decimal places. Consequently, the ratio of circumference to radius is 2π and the angle subtended at the centre of any circle by its circumference is 2π rad. But the complete angle at the centre of a circle is 360°, so we obtain the general relationship

Since 2π = 6.2832 (to four decimal places) it follows that 1 radian = 57.30°, as claimed earlier. Table 1 gives some angles measured in degrees and radians. As you can see from this table, many commonly–used angles are simple fractions or multiples of π radians, but note that angles expressed in radians are not always expressed in terms of π. Do not make the common mistake of thinking that π is some kind of angular unit; it is simply a number.

2.2 Right–angled triangles

Figure 4 shows a general triangle – a geometrical figure made up of three straight lines. There are three angles formed by the intersections of each pair of lines. By convention, the three angles are labelled according to the corresponding vertices as A^{^}, B^{^} and C^{^}. An alternative notation is to use AB^{^}C instead of B^{^}, where AB and BC are the two lines defining the angle. In this module we use the AB^{^}C and B^{^} notations interchangeably. i

The angles A^{^}, B^{^} and C^{^} in Figure 4 are known as the interior angles of the triangle.

The sum of the interior angles of any triangle is 180° (π radians). To see that this is so, imagine an arrow located at point A, superimposed on the line AB and pointing towards B. Rotate the arrow anticlockwise about the point A through an angle A^{^} so that it is aligned with AC and points towards C. Then rotate the arrow through an additional angle C^{^} so that it is horizontal and points to the right, parallel with BC. Finally, rotate the arrow through an additional angle B^{^} about point A. After this final rotation the arrow will again be aligned with the line AB, but will now point away from B having rotated through a total angle of 180°. So, rotating through A^{^} + B^{^} + C^{^} is equivalent to half a complete turn, i.e. 180°.

For the time being, we will be concerned only with the special class of triangles in which one interior angle is 90°. There are infinitely many such triangles, since the other two interior angles may have any values provided their sum is also 90°.

Any such triangle is called a right–angled triangle (see Figure 5), and the side opposite the right angle is known as the hypotenuse.

Notice the use of the box symbol, ⦜, to denote a right angle. Right–angled triangles are important because:

• There is a special relationship between the lengths of the sides (Pythagoras’s theorem.)
• There is a strong link between right–angled triangles and the trigonometric ratios (sine, cosine and tangent).
• Applications involving right–angled triangles often occur in physics (and many other branches of science and engineering).

We shall consider each of these points in this module, but the rest of this subsection will be devoted to Pythagoras’s theorem.

Consideration of Figure 6 shows how this result comes about for a general right–angled triangle of sides x, y and h. One way of finding the area of the large outer square is by squaring the length of its sides, i.e.

area = (x + y)² = x² + y² + 2xy(2)

However, the area of the large square can also be found by adding the area of the smaller square, h², to the areas of the four corner triangles. Each triangle has an area xy/2 (each is half a rectangle of sides x and y) so the area of the large square is

area = h² + 4xy/2 = h² + 2xy(3)

Comparing the right–hand sides of Equations 2 and 3 shows that

h² = x² + y²(4)

which is Pythagoras’s theorem.

The converse of Pythagoras’s theorem is also true; that is, if the sum of the squares of two sides of a triangle is equal to the square of the other side, then the triangle is right–angled. For the purpose of this module, we will accept the validity of the converse without proving it.

The sides of some right–angled triangles can be expressed entirely in terms of integers; probably the most famous is the 3 : 4 : 5 triangle where the hypotenuse has length 5 units and the other two sides have lengths 3 and 4 units: 3² + 4² = 5².

Note Throughout the remainder of this module we will not usually express lengths in any particular units. This is because we are generally interested only in the ratios of lengths. Of course, when you are considering real physical situations, you must attach appropriate units to lengths.

2.3 The trigonometric ratios

In the previous subsection, we indicated that the ratios of the lengths of the sides of a triangle were often of more interest than the actual lengths themselves. Figure 7 shows some similar triangles, i.e. triangles that are the same shape but different sizes – in other words, triangles with corresponding angles that are equal but with corresponding sides of different lengths. Although the lengths of the sides of any one triangle may differ from those of any similar triangle, the ratios of the side lengths are the same in each triangle – for example, each triangle in Figure 7 has sides whose lengths are in the ratio 2 : 3 : 4. In future, whenever we say that two or more triangles are similar we will mean it in the technical sense that they have the same interior angles and side lengths that are in the same ratio.

Figure 8 shows a right–angled triangle in which an angle θ has been marked for particular attention and the opposite side and adjacent side to this angle have been identified. Thus the three sides may be referred to as the opposite, the adjacent and the hypotenuse, and we may use these terms or the letters o, a and h to refer to their respective lengths.

Thanks to the special properties of right–angled triangles, the whole class of triangles that are similar to the triangle in Figure 8 can be characterized by the single angle θ, or, equivalently, by the ratio of the side lengths o : a : h. In fact, because of the Pythagorean relationship that exists between the sides of a right–angled triangle, the ratio of any two side lengths is sufficient to determine θ and identify the class of similar triangles to which a given right–angled triangle belongs. The ratios of the sides of right–angled triangles are therefore of particular importance.

The study of right–angled triangles is known as trigonometry, and the three distinct ratios of pairs of sides are collectively known as the trigonometric ratios. They are called the sine, cosine and tangent of the angle θ – abbreviated to sin, cosand tan, respectively i – and defined as follows:

The angle θ that appears in these definitions must lie between 0 rad (0°) and π/2 rad (90°), but later in this module we will extend the definitions to all angles. It should be emphasized that the value of a particular trigonometric ratio depends only on the value of θ, so the sine, cosine and tangent are effectively functions of the angle θ. More will be made of this point in Section 3 when we actually define related quantities called the trigonometric functions.

It is extremely useful to remember the definitions of the trigonometric ratios. You may find it helpful to denote the sine, cosine and tangent by the letters s, c and t and then, using h, o and a to represent hypotenuse, opposite and adjacent, the three relations read, left to right and top to bottom, as soh, cah and toa.

You can use a calculator to find the sine, cosine or tangent of an angle expressed in either degrees or radians, provided you first switch it to the appropriate mode – this is usually done by pressing a key marked ‘DRG’ (or something similar) until either ‘degrees’ or ‘radians’ appears in the display. Then key in the angle followed by one of the function keys sin, cosor tan.

The three trigonometric ratios are not all independent since we can write:

$\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} \times \dfrac{\text{hypotenuse}}{\text{adjacent}} =\sin\theta \times \dfrac{1}{\cos\theta}$

and so we obtain the identity:

Furthermore, we can use Pythagoras’s theorem to obtain a second relation. Starting with:

(opposite)² + (adjacent)² = (hypotenuse)²

we can divide both sides by (hypotenuse)² to obtain

$\dfrac{(\text{opposite})^2}{(\text{hypotenuse})^2} + \dfrac{(\text{adjacent})^2}{(\text{hypotenuse})^2}=1$

and therefore

(sinθ)² + (cosθ)² = 1

As you can see, writing powers of trigonometric functions can be rather cumbersome and so the convention that sinⁿθ means (sinθ)ⁿ (for positive values of n) is often used. Similar conventions are used for the other trigonometric functions. The notation cannot be used for negative values of n since sin⁻¹θ is sometimes used for the inverse sine function, which we consider later in this module. The above relation can therefore be written as:

Since there are two identities relating the trigonometric ratios, it follows that only one ratio is independent and therefore given one ratio we can find the other two. (This assumes that the trigonometric ratios are positive, which is true for 0° ≤ θ ≤ 90°.)

The angle ϕ in Figure 9 also has its sine, cosine and tangent. But the opposite and adjacent sides appropriate to θ are interchanged for ϕ and, as a consequence we can write

$\tan\theta = \dfrac{1}{\tan\phi},\;\sin\theta = \cos\phi,\;\cos\theta = \sin\phi$

There are some special angles for which it is easy to write down the sine, cosine and tangent.

As an example, consider the right–angled triangle with two sides of equal length, as shown in Figure 10.

Any triangle with two sides of equal length is called an isosceles triangle, and any isosceles triangle must contain two equal interior angles. The isosceles triangle of Figure 10 is special because it is also a right–angled triangle. Since the interior angles of any triangle add up to 180°, the angles of this particular triangle must be 45°, 90°, 45°. Also, since the two equal sides of this particular triangle are both of unit length it follows from Pythagoras’s theorem that the length of the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2\os}$ and so we can write down the following results:

$\sin 45° = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{1}{\sqrt{2\os}}$

$\cos 45° = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{1}{\sqrt{2\os}}$

$\tan 45° = \dfrac{\text{opposite}}{\text{adjacent}} = 1$

Figure 11 shows an equilateral triangle, i.e. one with three sides of equal length and hence three equal interior angles which must be equal to 60°. A line has been drawn from one vertex (i.e. corner) to the middle of the opposite side, so that the angle between the line and the side is 90° (that is, the line is a normal to the side).

One of the many reasons why trigonometric ratios i are of interest to physicists is that they make it possible to determine the lengths of all the sides of a right–angled triangle from a knowledge of just one side length and one interior angle (other than the right angle).

Here is an example of this procedure taken from the field of optics.

2.4 The reciprocal trigonometric ratios

The ratios introduced in the previous subsection could all have been written the other way up. The resulting reciprocal trigonometric ratios occur so frequently that they too are given specific names; they are the cosecant, secant, and cotangent (abbreviated to cosec, sec and cot) and are defined by:

Notice that cosec is the reciprocal of sin, and sec the reciprocal of cos. This terminology may seem rather odd but it is easily remembered by recalling that each reciprocal pair – (sin, cosec), (cos, sec), (tan, cot) – involves the letters ‘co’ just once. In other words there is just one ‘co’ between each pair. Also notice that each reciprocal trigonometric function is undefined when its partner function is zero.

Throughout the domains on which they are defined, each of the reciprocal trigonometric ratios can also be written in terms of the sides of the triangle in Figure 8:

Two useful general relationships follow from Equations 13–15 and Pythagoras’s theorem:

$\cot\theta = \dfrac{\cosec\,\theta}{\sec\theta}$(16)

and$1+\tan^2\theta=\sec^2\theta$(17)

Calculators do not generally have keys that give the reciprocal trigonometric ratios directly, but the ratios can be found using the sin, cosand tankeys and the reciprocal (1/x) key.

2.5 Small angle approximations

We end this section with some useful approximations involving small angles. Figure 15 shows a right–angled triangle with one very small angle θ and the third angle almost a right angle. If θ is at the centre of a circle radius r, where r is the hypotenuse of the triangle, you can see from the diagram that the opposite side to θ is almost coincident with the arc length s and the adjacent side to θ is almost the same length as the hypotenuse. From Equation 1, s/r is the value of θ in radians. So, for the small angle θ, Equations 5 to 7 give sinθ ≈ s/r, cosθ ≈ r/r, tanθ ≈ s/r and hence:

3 Graphs and trigonometric functions

3.1 The trigonometric functions

The ratio definitions of the sine, cosine and tangent (i.e. Equations 5, 6 and 7) only make sense for angles in the range 0 to π/2 radians, since they involve the sides of a right–angled triangle. In this subsection we will define three trigonometric functions, also called sine, cosine and tangent, and denoted sin(θ), cos(θ) and tan(θ), respectively. i These functions will enable us to attach a meaning to the sine and cosine of any angle, and to the tangent of any angle that is not an odd multiple π/2. Like the trigonometric ratios that they generalize, these trigonometric functions are of great importance in physics.

In defining the trigonometric functions we want to ensure that they will agree with the trigonometric ratios over the range 0 to π/2 radians. In order to do this, consider the point P shown in Figure 16 that moves on a circular path of unit radius around the origin O of a set of two–dimensional Cartesian coordinates (x, y).

If we let θ be the angle between the line OP and the x–axis, it follows from the definition of the trigonometric ratios that the coordinates of P are

y = sinθ      for 0 ≤ θ < π/2

andx = cosθ     for 0 ≤ θ < π/2

Although these trigonometric ratios are only defined over a very narrow range, it is easy to imagine the angle θ increasing in the positive (i.e. anticlockwise) direction to take up any positive value, with P crossing the positive x–axis whenever θ is equal to an integer multiple of 2π. And it is equally easy to imagine θ increasing in the negative (clockwise) direction to take up any negative value. Now, whatever the value of θ may be, large or small, positive or negative, the point P must still be located somewhere on the circle in Figure 16 and it must have a single x– and a single y–coordinate. We can use the particular values of x and y that correspond to a given value of θ to define the first two trigonometric functions

sin(θ) = y    for any θ

andcos(θ) = x    for any θ

Defined in this way, it is inevitable that the trigonometric functions will agree with the trigonometric ratios when 0 ≤ θ ≤ π/2, but it is also clear that the functions, unlike the ratios, make sense for arbitrary values of θ.

Having defined the functions sin(θ) and cos(θ) we also want to define a function tan(θ), but once again we want to ensure consistency with the behaviour of the trigonometric ratio tanθ that was introduced earlier. We can do this by using a generalization of Equation 8 (i.e. tanθ = sinθ1/cosθ0) as the basis of the definition. Thus, we define

$\tan(\theta)=\dfrac{\sin(\theta)}{\cos(\theta)} \quad\text{for any }\theta \ne(2n+1)\pi/2$ i

Notice that unlike the sine and cosine functions, this function is not defined for values of θ that are odd integer multiples of π/2. This restriction is imposed because cosθ = 0 at those values of θ, and the quotient sin(θ)/cos(θ) has no meaning. i

Having defined the trigonometric functions it is only natural to enquire about their graphs, since graphs are usually a good way of gaining insight into the behaviour of functions. Perhaps the first thing to notice about the trigonometric functions is that they are not always positive. If θ is in the range 0 ≤ θ < π/2 both the x– and y–coordinates of P will be positive, so both sin(θ) and cos(θ) will be positive as will their quotient tan(θ). However, as θ enters the range π/2 < θ < π the x–coordinate of P becomes negative, so cos(θ) and tan(θ) will be negative, though sin(θ) will remain positive. Similarly, when π < θ < 3π/2, x and y are both negative so sin(θ) and cos(θ) are negative while tan(θ) is positive; and if 3π/2 < θ < 2π, cos(θ) is positive while sin(θ) and tan(θ) are negative. As θ increases beyond 2π (or when θ decreases below 0) the same pattern is repeated.

Figure 17 summarizes the sign behaviour of the trigonometric functions. The positive function in each quadrant i is indicated by its initial letter, or in the case of the first quadrant where all the functions are positive by the letter A. Most people who use the trigonometric functions find it helpful to memorize Figure 17 (or to remember Figure 16, so that they can work it out). A traditional mnemonic to help recall which letter goes in which quadrant is ‘All Stations To Crewe’, which gives the letters in positive (anticlockwise) order starting from the first quadrant.

Of course, it is not only the signs of the trigonometric functions that change as θ increases or decreases and P moves around the circle in Figure 16. The values of x and y, and consequently of sin(θ), cos(θ) and tan(θ) also vary.

The graphs of sin(θ), cos(θ) and tan(θ) are shown in Figures 18, 19 and 20.

Table 3 The trigonometric functions between 0 and 2π.

θ/radians	sin θ	cos θ	tan θ
0	0	1	0
$\pi/6$	1/2	$\sqrt{3\os}/2$	$1/\sqrt{3\os}$
$\pi/4$	$1/\sqrt{2\os}$	$1/\sqrt{2\os}$	1
$\pi/3$	$\sqrt{3\os}/2$	1/2	$\sqrt{3\os}$
$\pi/2$	1	0	undefined
$2\pi/3$	$\sqrt{3\os}/2$	−1/2	−$\sqrt{3\os}$
$3\pi/4$	$1/\sqrt{2\os}$	−$1/\sqrt{2\os}$	−1
$5\pi/6$	1/2	−$\sqrt{3\os}/2$	−$1/\sqrt{3\os}$
$\pi$	0	−1	0
$7\pi/6$	−1/2	−$\sqrt{3\os}/2$	$1/\sqrt{3\os}$
$5\pi/4$	−$1/\sqrt{2\os}$	−$1/\sqrt{2\os}$	1
$4\pi/3$	−$\sqrt{3\os}/2$	−1/2	$\sqrt{3\os}$
$3\pi/2$	−1	0	undefined
$5\pi/3$	−$\sqrt{3\os}/2$	1/2	−$\sqrt{3\os}$
$7\pi/4$	−$1/\sqrt{2\os}$	$1/\sqrt{2\os}$	−1
$11\pi/6$	−1/2	$\sqrt{3\os}/2$	−$1/\sqrt{3\os}$
$2\pi$	0	1	0

As you can see, the values sin(θ) and cos(θ) are always in the range −1 to 1, and any given value is repeated each time θ increases or decreases by 2π.

The graph of tan(θ) (Figure 20) is quite different. Values of tan(θ) cover the full range of real numbers, but tan(θ) tends towards +∞ i as θ approaches odd multiples of π/2 from below, and towards −∞ as θ approaches odd multiples of π/2 from above. This emphasizes the impossibility of assigning a meaningful value to tan(θ) at odd multiples of π/2.

Table 3 lists the values of the trigonometric functions for some angles between 0 and 2π.

Given the trigonometric functions, we can also define three reciprocal trigonometric functions cosec(θ), sec(θ) and cot(θ), that generalize the reciprocal trigonometric ratios defined in Equations 10, 11 and 12.

$\cosec\,\theta = \dfrac{1}{\sin\theta} \quad\,\text{provided }\sin\theta \ne 0$(Eqn 10)

$\sec\theta = \dfrac{1}{\cos\theta} \quad\quad\text{provided }\cos\theta \ne 0$(Eqn 11)

$\cot \theta = \dfrac{1}{\tan\theta} \quad\quad\text{provided }\tan\theta \ne 0$(Eqn 12)

The definitions are straightforward, but a little care is needed in identifying the appropriate domain of definition in each case. (As usual we must choose the domain in such a way that we are not required to divide by zero at any value of θ.)

Graphs of the reciprocal trigonometric functions are shown in Figures 21, 22 and 23.

Throughout this subsection the argument θ of the various trigonometric and reciprocal trigonometric functions has always been an angle measured in radians. (This is true even though we have been conventionally careless about making sure that we always include the appropriate angular unit when assigning numerical values to θ.) However, the arguments of these functions do not have to be angles. If we regarded the numbers printed along the horizontal axes of Figures 18 to 23 as values of a purely numerical variable, x say, rather than values of θ in radians, we could regard the graphs as defining six functions of x; sin(x), cos(x), tan(x), etc. Strictly speaking these new functions are quite different from the trigonometric functions i and should be given different names to avoid confusion. But, given the tendency of physicists to be careless about domains and their habit of ‘dropping’ the explicit mention of radian from angular values, there is no practical difference between these new functions and the true trigonometric functions, so the confusion of names is harmless. Nonetheless, it is worth remembering that what appears as the argument of a trigonometric function is not necessarily an angle.

A common example of this arises in the study of oscillations i where trigonometric functions are used to describe repeated back and forth motion along a straight line.

In the simplest such motion, simple harmonic motion, the changing position x of a mass oscillating on the end of a spring may be represented by x = Acos(ωt + ϕ). Despite appearances none of the quantities inside the bracket is an angle (though they may be given angular interpretations); t is the time and is measured in seconds, ω is a constant known as the angular frequency that is related to the properties of the mass and spring and is measured in hertz (1 Hz = 1 s⁻¹), and ϕ, the phase constant, is a number, usually in the range 0 to 2π.

One final point to note. As mentioned before, throughout this subsection we have been careful to use brackets (as in sin(θ)) to distinguish the trigonometric functions from the trigonometric ratios (sinθ, etc)., but since the trigonometric functions and ratios agree in those regions where they are both defined this distinction is also of little importance in practice. Consequently, as a matter of convenience, the brackets are usually omitted from the trigonometric functions unless such an omission is likely to cause confusion. In much of what follows we too will omit them and simply write the trigonometric and reciprocal trigonometric functions as sinx, cosx, tanx, cosecx, secx and cot1x.

3.2 Periodicity and symmetry

The trigonometric functions are all examples of periodic functions. That is, as θ increases steadily, the same sets of values are ‘recycled’ many times over, always repeating exactly the same pattern. The graphs in Figures 18, 19 and 20, show this repetition, known as periodicity, clearly. More formally, a periodic function f(x) is one which satisfies the condition f(x) = f(x + nk) i for every integer n, where k is a constant, known as the period.

As noted in the answer to Question T12, the trigonometric functions have some symmetry either side of θ = 0. From Figures 18, 19 and 20 we can see the effect of changing the sign of θ:

Any function f(x) for which f(−x) = f(x) is said to be even_functioneven or symmetric_functionsymmetric, and will have a graph that is symmetrical about x = 0. Any function for which f(−x) = −f(x) is said to be odd_functionodd or antisymmetric_functionantisymmetric, and will have a graph in which the portion of the curve in the region x < 0 appears to have been obtained by reflecting the curve for x > 0 in the vertical axis and then reflecting the resulting curve in the horizontal axis. It follows from Equations 18, 19 and 20 that cos(θ) is an even function, while sin(θ) and tan(θ) are both odd functions.

The combination of periodicity with symmetry or antisymmetry leads to further relationships between the trigonometric functions. For example, from Figures 18 and 19 you should be able to see that the following relationships hold true:

sin(θ) = sin(π − θ) = −sin(π + θ) = −sin(2π−θ)(Eqn 21)

andcos(θ) = −cos(π − θ) = −cos(π + θ) = cos(2π − θ)(Eqn 22)

Less apparent, but just as true are the following relationships

cos(θ) = sin(π/2 − θ) = sin(π/2 + θ) = −sin(3π/2 − θ) = −sin(3π/2 + θ)(23)

andsin(θ) = cos(π/2 − θ) = −cos(π/2 + θ) = −cos(3π/2 − θ) = cos(3π/2 + θ)(24)

Thanks to periodicity, all of these relationships (Equations 21 to 24) remain true if we replace any of the occurrences of θ by (θ + 2nπ), where n is any integer.

It is quite clear from Figures 18 and 19 that there must be a simple relationship between the functions sinθ and cosθ0; the graphs have exactly the same shape, one is just shifted horizontally relative to the other through a distance π/2. Equations 23 and 24 provide several equivalent ways of describing this relationship algebraically, but perhaps the simplest is that given by the first and third terms of Equation 23:

i.e. sin(θ + π/2) = cos(θ)

Clearly, adding a positive constant, π/2, to the argument of the function has the effect of shifting the graph to the left by π/2. In crude terms, the addition has boosted the argument and makes everything happen earlier (i.e. further to the left).

Simple as it is, this is just one example of an important general principle that has many physical applications and deserves special emphasis.

Since the addition of a phase constant shifts a graph but does not change its shape, all graphs of sin(θ + ϕ) and cos(θ + ϕ) have the same ‘wavy’ shape, regardless of the value of ϕ: any function that gives a curve of this shape, or the curve itself, is said to be sinusoidal.

3.3 Inverse trigonometric functions

A problem that often arises in physics is that of finding an angle, θ, such that sinθ takes some particular numerical value. For example, given that sinθ = 0.5, what is θ? You may know that the answer to this specific question is θ = 30° (i.e. π/6); but how would you write the answer to the general question, what is the angle θ such that sinθ = x? The need to answer such questions leads us to define a set of inverse trigonometric functions that can ‘undo’ the effect of the trigonometric functions. These inverse functions are called arcsine, arccosine and arctangent (usually abbreviated to arcsin(x), arccos(x) and arctan(x)) and are defined so that:

Thus, since sin(π/6) = 0.5, we can write arcsin(0.5) = π/6 (i.e. 30°), and since tan(π/4) = 1, we can write arctan(1) = π/4 (i.e. 45°). Note that the argument of any inverse trigonometric function is just a number, whether we write it as x or sinθ or whatever, but the value of the inverse trigonometric function is always an angle. Indeed, an expression such as arcsin(x) can be crudely read as ‘the angle whose sine is x.’ Notice that Equations 25a–c involve some very precise restrictions on the values of θ, these are necessary to avoid ambiguity and deserve further discussion.

Looking back at Figures 18, 19 and 20, you should be able to see that a single value of sin(θ), cos(θ) or tan(θ) will correspond to an infinite number of different values of θ. For instance, sin(θ) = 0.5 corresponds to θ = π/6, 5π/6, 2π + (π/6), 2π + (5π/6), and any other value that may be obtained by adding an integer multiple of 2π to either of the first two values. To ensure that the inverse trigonometric functions are properly defined, we need to guarantee that each value of the function’s argument gives rise to a single value of the function. The restrictions given in Equations 25a–c do ensure this, but they are a little too restrictive to allow those equations to be used as general definitions of the inverse trigonometric functions since they prevent us from attaching any meaning to an expression such as arcsin(sin(7π/6)).

In fact, the general definitions of the inverse trigonometric functions are as follows:

The graphs of these three inverse trigonometric functions are given in Figure 25.

Equations 26a–c look more intimidating than Equations 25a–c, but they embody the same ideas and they have the advantage of assigning meaning to expressions such as arcsin(sin(7π/6)).

arcsin(sinθ) = θ    −π/2 ≤ θ ≤ π/2(Eqn 25a)

arccos(cosθ) = θ   0 ≤ θ ≤ π(Eqn 25b)

arctan(tanθ) = θ    −π/2 < θ < π/2(Eqn 25c)

If sin(θ) = x,  where −π/2 ≤ θ ≤ π/2  and  −1 ≤ x ≤ 1   then arcsin(x) = θ(Eqn 26a)

If cos(θ) = x,  where 0 ≤ θ ≤ π   and   −1 ≤ x ≤1   then arccos(x) = θ(Eqn 26b)

If tan(θ) = x,   where   −π/2 < θ < π/2   then   arctan(x) = θ(Eqn 26c)

Inverse trigonometric functions can be found on all ‘scientific’ calculators, often by using the trigonometric function keys in combination with the inverse key. The answer will be given in either radians or degrees, depending on the mode selected on the calculator, and will always be in the standard angular ranges given in Equations 26a–c.

When dealing with functions of any kind, physicists traditionally pay scant regard to such mathematical niceties as domains (ranges of allowed argument values) and codomains (ranges of allowed function values). This attitude persists because it rarely leads to error. However, care is needed when dealing with inverse trigonometric functions. If you know the sin, cosor tanof an angle and you want to find the angle itself, the inverse trigonometric functions will give you an answer. But that answer will lie in one of the standard angular ranges and may not be the particular answer you are seeking. You will have to remember that there are other possible answers, your calculator is unlikely to warn you!

It is possible to define a set of inverse reciprocal trigonometric functions in much the same way that we defined the inverse trigonometric functions. These functions are called arccosecant, arcsecant and arccotangent, and are usually abbreviated to arccosec, arcsecand arccot. Their definitions are:

The corresponding graphs are given in Figure 26.

Finally, a few words of warning. The following notation is sometimes used to represent the inverse trigonometric functions:

sin⁻¹(x) for arcsin(x),   cos⁻¹(x) for arccos(x)   and   tan⁻¹(x) for arctan(x)

Notice that there is no connection with the positive index notation used to denote powers of the trigonometric functions (for example, using sin²(θ) to represent (sin(θ))². Also notice that although this notation might make it appear otherwise, there is still a clear distinction between the inverse trigonometric functions and the reciprocal trigonometric functions.

Sosin⁻¹[sin(θ)] = θ

butcosec(θ) × sin(θ) = 1

4 Trigonometric rules and identities

In this section we first extend the discussion in Section 2 to derive relationships between sides and angles of any arbitrary triangle, and then add to the discussion of Subsection 2.3Subsections 2.3 and Subsection 3.23.2 by introducing some further relationships involving trigonometric functions.

Study comment It is unlikely that you will be required to reproduce the derivations given in this section. However, you should try to ensure that you understand each step, since that will reinforce your understanding and experience of trigonometric functions.

4.1 The sine and cosine rules

In Section 2, you saw that the sides and angles of a right–angled triangle are related by Pythagoras’s theorem and by the trigonometric ratios. These relationships are in fact special cases of relationships that apply to any triangle. To derive these general relationships, consider Figure 28, where ABC is any triangle, with sides a, b and c. BD is drawn perpendicular to AC and has length p. We have called the length of AD, q, which means that the length of DC is b − q.

From Figure 28

$\sin\hat {\rm A} = \dfrac pc \quad\text{so}\quad p = c \sin\hat {\rm A}$

$\sin\hat {\rm C} = \dfrac pa \quad\text{so}\quad p = a \sin\hat {\rm C}$

and therefore $a\sin\hat {\rm C} = c\sin\hat{\rm A}\quad\text{or}\quad\dfrac{a}{\sin\hat {\rm A}} = \dfrac{c}{\sin\hat {\rm C}}$

Similarly, by drawing a perpendicular from C to AB,

$\dfrac{a}{\sin\hat {\rm A}} = \dfrac{b}{\sin\hat {\rm B}}$

By combining the two previous equations we obtain

There is also an important relationship between the three sides of a general triangle and the cosine of one of the angles. i

To derive this rule, consider Figure 28 again. ABD is a right–angled triangle so Pythagoras’s theorem gives

c² = q² + p²

In a similar way, since BCD is a right–angled triangle, we have

a² = p² + ( b − q )²

Eliminating p² between these two simultaneous equations gives us:

a² = (c² − q²) + (b − q)²

which we can simplify by expanding the bracket on the right–hand side

a² = c² − q² + b² − 2bq + q² = c² + b² − 2bq

But q = ccosA^{^} and so

4.2 Trigonometric identities

A good deal of applicable mathematics is concerned with equations. It is generally the case that these equations are only true when the variables they contain take on certain specific values; for example, 4x = 4 is only true when x = 1. However, we sometimes write down equations that are true for all values of the variables, such as (x + 1)² = x² + 2x + 1. Equations of this latter type, i.e. ones that are true irrespective of the specific values of the variables they contain, are properly called identities. i

There are a great many trigonometric identities, i.e. relationships between trigonometric functions that are independent of the specific values of the variables they involve. These have various applications and it is useful to have a list of them for easy reference. The most important are given below – you have already met the first seven (in slightly different forms) earlier in the module and others occur at various points throughout FLAP. Note that α and β may represent any numbers or angular values, unless their values are restricted by the definitions of the functions concerned.

The symmetry relations: i

sin(−α) = −sin(α)(Eqn 18)

cos(−α) = cos(α)(Eqn 19)

tan(−α) = −tan(α)(Eqn 20)

The basic identities: i

$\tan(\alpha) = \dfrac{\sin(\alpha)}{\cos(\alpha)}$(Eqn 8)

sin²(α) + cos²(α) = 1(Eqn 9)

$\cot(\alpha) = \dfrac{\cosec\,(\alpha)}{\sec(\alpha)}$(Eqn 16)

$1+\tan^2(\alpha) = \sec^2(\alpha)~~~\text{for}~\alpha \ne (2n+1)\dfrac{\pi}{2}$(Eqn 17)

cot²(α) + 1 = cosec²(α)

The addition formulae: i

sin(α + β) = sin(α) cos(β) + cos(α) sin(β)(29)

cos(α + β) = cos(α) cos(β) − sin(α) sin(β)(30)

$\tan(\alpha+\beta) = \dfrac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)}$(31)

$\text{for}~\alpha,~\beta~\text{and}~ (\alpha+\beta) \ne (2n+1)\dfrac{\pi}{2}$

The double–angle formulae: i

sin(2α) = 2sin(α)cos(α)(32)

cos(α) = cos²(α) − sin²(α)(33)

cos(2α) = 1 − sin²(α)(34)

cos(2α) = 2cos²(α) − 1(35)

tan(2α) = 2tan(α)(36)

$\text{for}~\alpha~\text{and}~2\alpha \ne (2n+1)\dfrac{\pi}{2}$

The half–angle formulae: i

$\cos^2(\alpha/2) = \dfrac12[1 + \cos(\alpha)]$(37)

$\sin^2(\alpha/2) = \dfrac12[1 - \cos(\alpha)]$(38)

and if t = tan(α/2) then, provided tan(α/2) is defined:

$\sin(\alpha) = \dfrac{2t}{1+t^2}$(39)

$\cos(\alpha) = \dfrac{1-t^2}{1+t^2}$(40)

$\tan(\alpha) = \dfrac{2t}{1-t^2} \quad\text{for }\alpha \ne (2n+1)\dfrac{\pi}{2}$(41)

The sum formulae: i

$\sin(\alpha)+\sin(\beta) = 2\sin\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha-\beta}{2}\right)$(42)

$\sin(\alpha)-\sin(\beta) = 2\cos\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha-\beta}{2}\right)$(43)

$\cos(\alpha)+\cos(\beta) = 2\cos\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha-\beta}{2}\right)$(44)

$\cos(\alpha)-\cos(\beta) = -2\sin\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha-\beta}{2}\right)$(45)

The product formulae: i

2sin(α)cos(β) = sin(α + β) + sin(α − β)(46)

2cos(α)sin(β) = sin(α + β) − sin(α − β)(47)

2cos(α)cos(β) = cos(α + β) + cos(α − β)(48)

−2sin(α)sin(β) = cos(α + β) − cos(α − β)(49)

As indicated by the marginal notes, some of these identities can be derived straightforwardly from others. Rather than deriving the unexplained identities, one by one, we will consider just one example in full and allow it to serve as an example.

To derive Equation 29, the addition formula for sin(α + β), we use Figure 31.

sin(α + β) = sin(α) cos(β) + cos(α) sin(β)(Eqn 29)

The line OB is at an angle, α, to the x–axis. OA has unit length and is at an angle, β, to OB. The angle AB^{^}O is a right angle. (This is what fixes the length of OB, which is not of unit length.) It turns out to be useful to construct lines AN, BM and TB, such that AN^{^}O, BM^{^}O and BT^{^}A, are all right angles. If we let S represent the point of intersection of OB and AN, then OS^{^}N, and consequently AS^{^}B, are equal to α − 90°. But SAB is a right–angled triangle, so TA^{^}B is equal to α.

Since OA has unit length, the y–coordinate of the point, A, is sin(α + β) and from the diagram we have:

sin(α + β) = NT + TA(50)

To find TA, first consider the triangle, OAB. Since OA = 1, we have:

AB = sin(β)(51)

Next, considering triangle TAB, we have

$\cos(\alpha) = \dfrac{\rm{TA}}{\rm{AB}} = \dfrac{\rm{TA}}{\sin\beta}$(52)

Therefore,

TA = cos(α)sin(β)(53)

To find NT, we start by considering triangle OAB, obtaining

OB = cos(β)(54)

Next, consideration of triangle OBM gives

$\sin(\alpha) = \dfrac{\rm{MB}}{\rm{OB}} = \dfrac{\rm{NT}}{\cos\beta}$(55)

Therefore,

NT = sin(α)cos(β)(56)

We can now substitute in Equation 50,

sin(α + β) = NT + TA(Eqn 50)

using Equations 53 and 56 to get the desired result:

TA = cos(α)sin(β)(Eqn 53)

NT = sin(α)cos(β)(Eqn 56)

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)(Eqn 29) i

The first of the double–angle identities, Equation 32, can be obtained by putting β = α in Equation 29: i

sin(α + α) = 2sin(α)cos(α)

Therefore,

sin(2α) = 2sin(α)cos(α)(Eqn 32)

5 Closing items

5.1 Module summary

1

Angles are usually measured in Subsection 2.1degrees or Subsection 2.1radians. An angle of 180° is equivalent to π rad.

2

The side opposite the Subsection 2.2right angle in a right–angled triangle is called the hypotenuse.

3

Pythagoras’s theorem states that the square of the hypotenuse in a right–angled triangle is the sum of the squares of the other two sides. (The converse is also true.)

4

For a right–angled triangle, the Subsection 2.3trigonometric ratios are:

$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$(Eqn 5)

$\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$(Eqn6)

$\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$(Eqn 7) i

5

The Subsection 2.4reciprocal trigonometric ratios are:

$\cosec\,\theta = \dfrac{1}{\sin\theta} \quad\,\text{provided }\sin\theta \ne 0$(Eqn 10)

$\sec\theta = \dfrac{1}{\cos\theta} \quad\quad\text{provided }\cos\theta \ne 0$(Eqn 11)

$\cot \theta = \dfrac{1}{\tan\theta} \quad\quad\text{provided }\tan\theta \ne 0$(Eqn 12)

6

For a small angle θ, cosθ ≈ 1

and sinθ ≈ tanθ ≈ θ/rad

7

The sides and angles of a general triangle are related by

the Subsection 4.1sine rule:

$\dfrac{a}{\sin\hat A}=\dfrac{b}{\sin\hat B}=\dfrac{c}{\sin\hat C}$(Eqn 27)

and the cosine rule:

$a^2 = b^2 + c^2- 2bc\cos\hat A$(Eqn 28)

8

The Subsection 3.1trigonometric functions generalize the trigonometric ratios; sin(θ) and cos(θ) are periodic functions (with period 2π) and are defined for any value of θ. The function tan(θ) is a periodic function (with period π) and is defined for all values of θ except odd multiples of π/2. The reciprocal trigonometric functions generalize the reciprocal trigonometric ratios in a similar way.

9

The Subsection 3.3inverse trigonometric functions; arcsin(x), arccos(x) and arctan(x) are defined as follows:

If sin(θ) = x,  where −π/2 ≤ θ ≤ π/2  and  −1 ≤ x ≤ 1    then arcsin(x) = θ(Eqn 26a)

If cos(θ) = x,  where 0 ≤ θ ≤ π   and   −1 ≤ x ≤1    then arccos(x) = θ(Eqn 26b)

If tan(θ) = x,  where −π/2 < θ < π/2    then arctan(x) = θ(Eqn 26c)

The inverse reciprocal trigonometric functions may be defined in a similar way.

10

There are a large number of Subsection 4.2trigonometric identities, such as

sin²(α) + cos²(α) = 1(Eqn 9)

The most important are listed in Subsection 4.2 of this module.

5.2 Achievements

Having completed this module, you should be able to:

A1

Define the terms that are emboldened and flagged in the margins of the module.

A2

Given an angle measured in degrees express its value in radians, and vice versa.

A3

Use Pythagoras’s theorem.

A4

Use trigonometric ratios, and their reciprocals, to solve geometrical problems.

A5

Explain the sine and cosine rules and use these rules to solve problems involving general triangles.

A6

Explain how the sine, cosine and tangent functions can be defined for general angles and sketch their graphs.

A7

Recognize the most common identities for the trigonometric functions and apply them to solving mathematical and physical problems.

A8

Use the inverse trigonometric functions to solve mathematical and physical problems.

Study comment You may now wish to take the following Exit test for this module which tests these Achievements. If you prefer to study the module further before taking this test then return to the topModule contents to review some of the topics.

5.3 Exit test

Study comment Having completed this module, you should be able to answer the following questions, each of which tests one or more of the Achievements.

FLAP material © copyright 1996 Open University