90 Chapter 3Transcendental functions
Section 3.3
12.Find the principal values of
(i) (ii)sin
− 1
(1) (iii) (iv)cos
− 1
(−1)
13.The Bragg equation for the reflection of radiation of wavelength λfrom the planes of a
crystal isnλ 1 = 12 d 1 sinθwhere dis the separation of the planes, θis the angle of incidence
of the radiation, and nis an integer. Calculate the angles θat which X-rays of wavelength
1.5 1 × 110
− 10
m are reflected by planes separated by3.0 1 × 110
− 10
m.
Section 3.4
14.Given the sidea 1 = 11 and anglesA 1 = 1 π 24 andB 1 = 1 π 23
of a triangle ABC, Figure 3.24, find the third angle and
the other two sides.
15.Given the sidesa 1 = 12 ,b 1 = 1 2.5andc 1 = 13 of a triangle
ABC, find the angles.
16.Given the sidesa 1 = 13 , b 1 = 14 , and included angle
C 1 = 1 π 24 of triangle ABC, find the third side and the
other two angles.
17.Given the sides , and included angleC 1 = 1 π 24 of the triangle ABC, find the
third side and the other two angles.
18.Express in terms of in terms of the sines and cosines of 2 θand 5 θ:
(i)sin 17 θ, (ii)sin 13 θ, (iii)cos 17 θ, (iv)cos 13 θ.
19.Express (i)sin 13 θin terms of sin 1 θ, (ii)cos 13 θin terms ofcos 1 θ.
20.Expresscos 14 xin terms of
(i)sin 12 xand cos 12 x, (ii)sin 12 xonly, (iii)cos 12 xonly, (iv)sin 1 xonly,
(v)cos 1 xonly.
21.Givensin 1 10° 1 = 1 0.1736, sin 1 30° 1 = 1122 , sin 1 50° 1 = 1 0.7660, findcos 1 20°(without using a
calculator).
22.Express in terms of the sines of 8xand 2x: (i)sin 15 x 1 cos 13 x, (ii)cos 15 x 1 sin 13 x.
23.Express in terms of the cosines of 8xand 2x: (i)sin 15 x 1 sin 13 x, (ii)cos 15 x 1 cos 13 x.
24.Express (i)sin(π 1 ± 1 θ)and (ii)cos(π 1 ± 1 θ)in terms ofsin 1 θandcos 1 θ.
25.The function ψ(x, t) 1 = 1 sin 1 πx 1 cos 12 πtrepresents a standing wave. Find the values of time t
for whichψhas (i)maximum amplitude, (ii)zero amplitude. (iii)Sketch the wave
function betweenx 1 = 10 andx 1 = 13 at (a) t 1 = 10 , (b) t 1 = 1128.
26.The function
represents the superposition of two harmonic waves with the same wavelength λ. Show
that φis (i)also harmonic with the same wavelength, and (ii)can be written as
where and tan 1 α 1 = 1 b 2 a.
Section 3.5
27.Find the cartesian coordinates of the points whose polar coordinates are
(i)r 1 = 1 3, θ 1 = 1 π 2 3, (ii)r 1 = 1 3, θ 1 = 15 π 2 3.
Aab=+
22
φ
λ
() sinxA α
x
=+
2 π
φ
λλ
() sinxa cos
x
b
x
=+
22 ππ
ab=,= 23
cos
−
()
1 1
2
sin
−
()
11
2
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
....
...
...
...
...
...
....
...
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
...
....
...
...
...
...
...
....
...
...
...
...
...
...
...
...
....
...
...
..A
B
C
a
b
c
...................
...
...
..
...
..
...
......
............................Figure 3.24