sin A + sin B = 2 sin A + B 2
cos A − B 2
Solution
Consider the identitiessin(C + D) = sin C cos D + sin D cos C sin(C − D) = sin C cos D − sin D cos C By adding these identities we obtain
sin(C + D) + sin(C − D) = 2 sin C cos D
We now make the substitutions C + D = A, C − D = B from which C = A + B
2 , D = A − B
2 Hence
sin A + sin B = 2 sin A + B 2
cos A − B 2
The result of Example 3.7 is one of many similar results. These are listed in Table 3.2. Table 3.2
Further trigonometric identities sin A + sin B = 2 sin A + B
2
cos A − B 2
sin A − sin B = 2 sin A − B 2
cos A + B 2
cos A + cos B = 2 cos A + B 2
cos A − B 2
cos A − cos B = −2 sin A + B 2 sin A − B 2
Example 3.8
Simplify sin 70◦−sin 30◦ cos 50◦Solution
We note that the numerator, sin 70◦ −sin 30◦, has the form sin A − sin B. Using theidentity for sin A − sin B with A = 70◦and B = 30◦we see
sin 70◦−sin 30◦=2 sin
70◦−30◦ 2 cos 70◦+30◦ 2 =2 sin 20◦cos 50◦
Hence sin 70◦−sin 30◦ cos 50◦ = 2 sin 20◦cos 50◦ cos 50◦ =2 sin 20◦
EXERCISES 3.6
1 Use the identities for sin(A ± B), cos(A ± B) and tan(A ± B) to simplify the following:
(a) sin θ −π 2 ! (b) cos θ −π 2 ! (c) tan(θ + π) (d) sin(θ − π) (e) cos(θ − π) (f) tan(θ − 3π) (g) sin(θ + π) (h) cos θ +3π 2 ! (i) sin 2 θ +3π 2 ! (j) cos θ −3π 2 ! (k) cos π 2+θ !
2 Write down the trigonometric identity for tan(A + θ ). By letting A →π
2 show that tan π 2 +θ
! can be simplified to − cot θ .
3 (a) By dividing the identity
sin2A + cos2A = 1 by cos2A show that
tan2A + 1 = sec2A.
(b) By dividing the identity
sin2A + cos2A = 1 by sin2A show that
1 + cot2A = cosec2A.
4 Simplify the following expressions: (a) cos A tan A (b) sin θ cot θ
(c) tan B cosec B (d) cot 2x sec 2x (e) tan θ tan π
2+θ !
(f) sin 2t cos t [Hint: see Question 2.]
(g) sin2A + 2 cos2A (h) 2 cos2B − 1
(i) (1 + cot2X ) tan2X (j) (sin2A + cos2A)2
(k) 1
2sin 2A tan A (l) (sec2t − 1) cos2t (m) sin 2A
cos 2A (n)
sin A sin 2A
(o) (tan2θ +1) cot2θ (p) cos 2A + 2 sin2A
5 Simplify
(a) sin 110◦−sin 70◦
(b) cos 20◦−cos 80◦ (c) sin 40◦+sin 20◦ (d) cos 50◦√+cos 40◦ 2 6 Show that sin 60◦+sin 30◦ sin 50◦−sin 40◦ is equivalent to cos 15◦ sin 5◦
Solutions
1 (a) − cos θ (b) sin θ (c) tan θ (d) − sin θ (e) − cos θ (f) tan θ (g) − sin θ (h) sin θ (i) − cos 2θ (j) − sin θ (k) − sin θ
4 (a) sin A (b) cos θ (c) sec B (d) cosec 2x (e) −1 (f) 2 sin t
(g) 1 + cos2A (h) cos 2B (i) sec2X
(j) 1 (k) sin2A (l) sin2t
(m) tan 2A (n) 12sec A (o) cosec2θ (p) 1
3.7 Modelling waves using sint and cos t 131
3.7
MODELLING WAVES USING SINt AND COS t
Examining the graphs of sin x and cos x reveals that they have a similar shape to waves. In fact, sine and cosine functions are often used to model waves and we will see in Chapter 23 that almost any wave can be broken down into a combination of sine and cosine functions. The main waves found in engineering are ones that vary with time and so t is often the independent variable.
The amplitude of a wave is the maximum displacement of the wave from its mean position. So, for example, sin t and cos t have an amplitude of 1, the amplitude of 2 sin t is 2, and the amplitude of A sin t is A (see Figure 3.15).
The amplitude of A sin t is A. The amplitude of A cos t is A.
A more general wave is defined by A cos ωt or A sin ωt. The symbol ω represents the
angular frequency of the wave. It is measured in radians per second. For example, sin 3t
has an angular frequency of 3 rad s−1. As t increases by 1 second the angle, 3t, increases
by 3 radians. Note that sin t has an angular frequency of 1 rad s−1.
The angular frequency of y = A sin ωt and y = A cos ωt is ω radians per second. The sine and cosine functions repeat themselves at regular intervals and so are pe-
riodic functions. Looking at Figure 3.7 we see that one complete cycle of sin t is com-
pleted every 2π seconds. The time taken to complete one full cycle is called the period and is denoted by T . Hence the period of y = sin t is 2π seconds. Similarly the period of y = cos t is 2π seconds. Mathematically this means that adding or subtracting multiples of 2π to t does not change the sine or cosine of that angle.
sin t = sin(t ± 2nπ) n = 0, 1, 2, 3, . . . cos t = cos(t ± 2nπ) n = 0, 1, 2, 3, . . .
In particular we note that sin t = sin(t + 2π) cos t = cos(t + 2π)
We now consider y = A sin ωt and y = A cos ωt. When t = 0 seconds, ωt = 0 radians. When t = 2π
ω seconds, ωt = ω 2π
ω =2π radians. We can see that as t increases from 0
A sin t 2 sin t f (t) A –A 2 –2 2p t Figure 3.15
to2π
ω seconds, the angle ωt increases from 0 to 2π radians. We know that as the angle ωt increases by 2π radians then A sin ωt completes a full cycle. Hence a full cycle is completed in2π
ω seconds, that is the period of y = A sin ωt is 2π
ω seconds. If y = A sin ωt or y = A cos ωt, then the period T is2π
ω.
In particular we note that the period of y = A sin t and y = A cos t is 2π.
Closely related to the period is the frequency of a wave. The frequency is the number of cycles completed in 1 second. Frequency is measured in units called hertz (Hz). One hertz is one cycle per second. We have seen that y = A sin ωt takes
2π
ω seconds to complete one cycle and so it will take
1 second to complete ω 2πcycles
We use f as the symbol for frequency and so frequency, f = ω
2π
For example, sin 3t has a frequency of 3 2π
Hz. Note that by rearrangement we may write
ω =2π f
and so the wave y = A sin ωt may also be written as y = A sin 2π f t. From the definitions of period and frequency we can see that
period = 1 frequency that is
T = 1 f
We see that the period is the reciprocal of the frequency. Identical results apply for the wave y = A cos ωt.
A final generalization is to introduce a phase angle or phase, φ. This allows the wave to be shifted along the time axis. It also means that either a sine function or a cosine function can be used to represent the same wave. So the general forms are
A cos(ωt + φ), A sin(ωt + φ)
Figure 3.16 depicts A sin(ωt + φ). Note from Figure 3.16 that the actual movement of the wave along the time axis is φ/ω. It is easy to show this mathematically:
A sin(ωt + φ) = A sin ω t + φ ω The quantity φ
3.7 Modelling waves using sint and cos t 133 f (t) A t 1 – f 2p– v f – v T = = Figure 3.16
The generalized wave A sin(ωt + φ). The waves met in engineering are often termed signals or waveforms. There are no rigid rules concerning the use of the words wave, signal and waveform, and often engineers use them interchangeably. We will follow this convention.
Example 3.9
State the amplitude, angular frequency and period of each of the following waves: (a) 2 sin 3t (b) 1 2cos 2 t + π 6Solution
(a) Amplitude, A = 2, angular frequency, ω = 3, period, T = 2π ω =2π 3 . (b) Amplitude, A = 0.5, angular frequency, ω = 2, period, T = 2π
ω = 2π
2 = π.
Example 3.10
State the amplitude, period, phase angle and time displacement of (a) 2 sin(4t + 1) (b) 2 cos(t − 0.7) 3 (c) 4 cos 2t + 1 3 (d) 3 4sin 4t 3Solution
(a) Amplitude = 2, period = 2π 4 =π
2, phase angle = 1 relative to 2 sin 4t, time displacement = 0.25.
(b) Amplitude = 2
3, period = 2π, phase angle = −0.7 relative to 2
3cos t, time displace- ment = −0.7. (c) Amplitude = 4, period = 2π 2/3 =3π, phase angle = 1 3 relative to 4 cos 2t 3 , time displacement = 0.5. (d) Amplitude = 3 4, period = 2π 4/3 = 3π
2 , phase angle = 0 relative to 3 4sin 4t 3 , time displacement = 0.