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IV. LA CUESTION SEMANTICA: «EXTENSION»
2. SOBRE LA EVOLUCION DE LA PRAXIS EXTENSIONISTALA PRAXIS EXTENSIONISTA
The static approach began with practical problems of surveying and gave rise to the mathematical problems of triangles and their measurement that we call trigonometry.
We consider a right-angled triangle ABC, where ∠CAB is the right-angle, and define the sine, cosine and tangent functions in relation to that triangle. Thus in Figure 2.47 we have
The way in which these functions were defined led to their being called the ‘trigono-metrical ratios’. The context of the applications implied that the angles were measured in the sexagesimal system (degees, minutes, seconds): for example 35°21′41″ which today is written in the decimal form 35.36°. In modern textbooks this is shown explicitly, writing, for example, sin 30°, or cos 35.36°, or tanθ°, so that the independent variable θ is a pure number. For example, by considering the triangles shown in Figure 2.48(a), we can readily write down the trigonometric ratios for 30°, 45° and 60° as indicated in the table of Figure 2.48(b).
tangent tan opposite
adjacent θ° = θ° = c =
b
cosine cos adjacent
hypotenuse θ° = θ° = b =
a
sine sin opposite
hypotenuse θ° = θ° = c =
a Figure 2.46
(a) Hipparchus: chords as a function of angle, expressed as parts of a radius. (b) Aryabhata:
half-chords as a function of angle, expressed as parts of the arc subtended by the angle with π 31 416/10 000.
Figure 2.47
To extend trigonometry to problems involving triangles that are not necessarily right-angled, we make use of the sine and cosine rules. Using the notation of Figure 2.49 (note that it is usual to label the side opposite an angle by the corresponding lower-case letter), we have, for any triangle ABC:
The sine rule
(2.15)
The cosine rule
a2= b2+ c2− 2bc cos A (2.16)
or
b2= a2+ c2− 2ac cos B or
c2= a2+ b2− 2ab cos C
Example 2.40 Consider the surveying problem illustrated in Figure 2.50. The height of the tower is to be determined using the data measured at two points A and B, which are 20 m apart.
The angles of elevation at A and B are 28°53′ and 48°51′ respectively.
a A
b B
c C sin = sin = sin Figure 2.48
Figure 2.49
Figure 2.50 Tower of Example 2.40.
Solution By elementary geometry
∠ACB = 40°51′ − 28°53′ = 11°58′
Using the sine rule, we have
= so that
CB= 20 sin(28°53′)/sin(11°58′) The height required CD is given by
CD= CB sin(48°51′)
= 20 sin(28°53′) × sin(48°51′)/sin(11°58′)
= 30.475
Hence the height of the tower is 30.5 m.
2.6.2 Exercises
AB sin(11°58′) CB
sin(28°53′)
47 In the triangles shown in Figure 2.51, calculate sinθ°, cos θ° and tan θ°. Use a calculator to determine the value of θ in each case.
Figure 2.51
48 In the triangle ABC shown in Figure 2.52, calculate the lengths of the sides AB and BC.
Figure 2.52
Figure 2.53 Optical angle of mural of Question 53.
50 Calculate the value of θ where cosθ° = 2 cos230°− 1
51 In triangle ABC, angle A is 40°, angle B is 60°
and side BC is 20 mm. Calculate the lengths of the remaining two sides.
52 In triangle ABC, the angle C is 35° and the sides AC and BC have lengths 42 mm and 73 mm respectively.
Calculate the length of the third side AB.
53 The lower edge of a mural, which is 4 m high, is 2 m above an observer’s eye level, as shown in Figure 2.53. Show that the optical angle θ° is given by
where dm is the distance of the observer from the mural. See Review exercises Question 23.
cos
[( )( )]
θ° = +
+ +
12
4 36
2
2 2
d
d d
÷
49 Calculate the value of θ where
sinθ° = sin 10° cos 20° + cos 10° sin 20°
2.6.3 Circular functions
The dynamic definition of the functions arises from considering the motion of a point P around a circle as shown in Figure 2.54. Many practical mechanisms involve this mathematical model.
The distance OP is one unit, and the perpendicular distance NP of P from the initial position OP0of the rotating radius is the sine of the angle ∠P0OP. Note that we are measuring NP positive when P is above OP0 and negative when P is below OP0. Similarly, the distance ON defines the cosine of ∠P0OP as being positive when N is to the right of O and negative when it is to the left of O.
Because we are concerned with circles and rotations in these definitions, it is natural to use circular measure so that ∠P0OP, which we denote by x, is measured in radians.
In this case we write simply sin x or cos x, where, as before, x is a pure number. One radian is the angle that, in the notation of Figure 2.54, is subtended at the centre when the arclength P0P is equal to the radius OP0. Obviously therefore
180°=π radians
a result we can use to convert degrees to radians and vice versa. It also follows from the definition of a radian that
(a) the length of the arc AB shown in Figure 2.55(a), of a circle of radius r, subtending an angle θ radians at the centre of the circle, is given by
length of arc = rθ (2.17)
(b) the area of the sector OAB of a circle of radius r, subtending an angle θ radians at the centre of the circle (shown shaded in Figure 2.55(b)), is given by
area of sector =12r2θ (2.18)
Figure 2.54
Figure 2.55 (a) Arc of a circle.
(b) Sector of a circle.
To obtain the graph of sin x, we simply need to read off the values of PN as the point P moves around the circle, thus generating the graph of Figure 2.56. Note that as we continue around the circle for a second revolution (that is, as x goes from 2π to 4π) the graph produced is a replica of that produced as x goes from 0 to 2π, the same being true for subsequent intervals of 2π. By allowing P to rotate clockwise around the circle, we see that sin(−x) = −sin x, so that the graph of sin x can be extended to negative values of x, as shown in Figure 2.57.
Since the graph replicates itself for every interval of 2π,
sin(x+ 2πk) = sin x, k = 0, ±1, ±2, . . . (2.19)
and the function sin x is said to be periodic with period 2π.
To obtain the graph of y= cos x, we need to read off the value of ON as the point P moves around the circle. To make the plotting of the graph easier, we first rotate the circle through 90° anticlockwise and then proceed as for y= sin x to produce the graph of Figure 2.58. By allowing P to rotate clockwise around the circle, we see that cos(−x) = cos x, so that the graph can be extended to negative values of x, as shown in Figure 2.59.
Figure 2.56 Generating the graph of sin x.
Figure 2.57 Graph of y= sin x.
Figure 2.58 Generating the graph of cos x.
Figure 2.59 Graph of y= cos x.
Again, the function cos x is periodic with period 2π, so that
cos(x+ 2πk) = cos x, k = 0, ±1, ±2, … (2.20)
Note also that the graph of y= sin x is that of y = cos x moved π units to the right, while that of y= cos x is the graph of y = sin x moved π units to the left. Thus, from Section 2.2.3,
sin x= cos(x − π) (2.21)
or
cos x= sin(x + 12π)
1 2
1 2
1 2
The definition of tan x is similar, and makes obvious the origin of the name ‘tangent’
for this function. In Figure 2.60 the rotating radius OP is extended until it cuts the tan-gent P0M to the circle at the initial position P0. The length P0M is the tangent of ∠P0OP.
Allowing P to move around the circle, we generate the graph shown in Figure 2.60.
Again, by allowing P to move in a clockwise direction, we have tan(−x) = −tan x, and the graph can readily be extended to negative values of x. In this case the graph replicates itself every interval of duration π so that
tan(x+πk) = tan x, k = 0, ±1, ±2, … (2.22)
and tan x is of period π.
These definitions of sine, cosine and tangent show how they are associated with the properties of the circle, and consequently they are called circular functions. Often in an engineering context, the static and dynamic uses of these functions occur simultan-eously. Consequently, we often refer to them as trigonometric functions.
Using the results (2.19), (2.20) and (2.22), it is possible to calculate the values of the trigonometric functions for angles greater than π using their values for angles between zero and π. The rule is: take the acute angle that the direction makes with the initial direction, find the sine, cosine or tangent of this angle and multiply by +1 or −1 according to the scheme of Figure 2.61. For example
cos(135°) = cos(180° − 45°) = −cos 45° = −÷
sin(330°) = sin(360° − 30°) = −sin 30° = − tan(240°) = tan(180° + 60°) = tan 60° = ÷3
1 2
1 2 1
2
1 2
Figure 2.60
Figure 2.61
As we frequently move between measuring angles in degrees and in radians, it is important to check that your calculator is in the correct mode.
If the radius OP is rotating with constant angular velocity ω (in rad s−1) about O then x=ωt, where t is the time (in s). The time T taken for one complete revolution is given by ωT = 2π; that is, T = 2π/ω. This is the period of the motion. In one second the radius makes ω/2π such revolutions. This is the frequency, ν. Its value is given by
Thus, the function y= A sinωt, which is associated with oscillatory motion in engineer-ing, has period 2π/ω and amplitude A. The term amplitude is used to indicate the maximum distance of the graph of y= A sinωt from the horizontal axis.
Example 2.41 Sketch using the same set of axes the graphs of the functions (a) y= 2 sin t (b) y= sin t (c) y= sin t
and discuss.
Solution The graphs of the three functions are shown in Figure 2.62. The functions (a), (b)and (c)have amplitudes 2, 1 and respectively. We note that the effect of changing the amplitude is to alter the size of the ‘humps’ in the sine wave. Note that changing only the amplitude does not alter the points at which the graph crosses the x axis. All three functions have period 2π.
1 2
1 2
ν =frequency = = period
1 2
ω π
Figure 2.62
Example 2.42 Sketch using the same axes the graphs of the functions (a) y= sin t (b) y= sin 2t (c) y= sin t and discuss.
Solution The graphs of the three functions (a), (b)and (c)are shown in Figure 2.63. All three have amplitude 1 and periods 2π, π and 4π respectively. We note that the effect of changing the parameter ω in sin ωt is to ‘squash’ or ‘stretch’ the basic sine wave sin t.
All that happens is that the basic pattern repeats itself less or more frequently; that is, the period changes.
1 2
In engineering we frequently encounter the sinusoidal function
y= A sin(ωt + α), ω 0 (2.23)
Following the discussion in Section 2.2.4, we have that the graph of this function is obtained by moving the graph of y= A sinωt horizontally:
units to the left if α is positive or
units to the right if α is negative
The sine wave of (2.23) is said to ‘lead’ the sine wave A sinωt when α is positive and to ‘lag’ it when α is negative.
Example 2.43 Sketch the graph of y= 3 sin(2t + π).
Solution First we sketch the graph of y = 3 sin 2t, which has amplitude 3 and period π, as shown in Figure 2.64(a). In this case α = π and ω = 2, so it follows that the graph of y = 3 sin(2t + π) is obtained by moving the graph of y = 3 sin 2t horizontally to the left by 16π units. This is shown in Figure 2.64(b).
1 3
1 3 1
3
| |α ω α ω Figure 2.63
Figure 2.64
Example 2.44 Consider the crank and connecting rod mechanism illustrated in Figure 2.65. Determine a functional relationship between the displacement of Q and the angle through which the crank OP has turned.
Solution As the crank OP rotates about O, the other end of the connecting rod moves backwards and forwards along the slide AB. The displacement of Q from its initial position depends on the angle through which the crank OP has turned. A mathematical model for the mechanism replaces the crank and connecting rod, which have thickness as well as length, by straight lines, which have length only, and we consider the motion of the point Q as the line OP rotates about O, with PQ fixed in length and Q constrained to move on the line AB, as shown in Figure 2.66. We can specify the dependence of Q on the angle of rotation of OP by using some elementary trigonometry. Labelling the length of OP as r units, the length of PQ as l units, the length of OQ as y units and the angle ∠AOP as x radians, and applying the cosine formula gives
Figure 2.65
Crank and connecting rod mechanism.
Figure 2.66 Model of crank and connecting rod.
l2= r2+ y2− 2yr cos x which implies
( y− r cos x)2 = l2− r2+ r2cos2x
= l2− r2sin2x and
y= r cos x + ÷(l2− r2sin2x)
Thus for any angle x we can calculate the corresponding value of y. We can represent this relationship by means of a graph, as shown in Figure 2.67.
Figure 2.67
In MATLAB the circular functions are represented by sin(x), cos(x) and tan(x) respectively. (Note that MATLAB uses radians in function evaluation.) Also in MATLAB pi (Pi in MAPLE) is a predefined variable representing the quantity π. As an example check that the commands
t = –2*pi : pi/90 : 2*pi;
y1 = sin(t); y2 = sin(2*t); y3 = sin(0.5*t);
plot(t, y1, ‘-’,t ,y2, ‘- -’, t, y3, ‘-.’) output the basic plots of Figure 2.63.
In symbolic form graphs may be produced using the ezplotcommand. Check that the commands
syms t
y = sym(3*sin(2*t + pi/3));
ezplot(y,[–2*pi,2*pi] ) grid
produce the plot of Figure 2.64(b).