Assign-05-H8152.tex 23/6/2006 15: 8 Page 189
B
Geometry and Trigonometry
Assignment 5
This assignment covers the material contained
in Chapters 15 to 18.The marks for each question are shown in
brackets at the end of each question.- Solve the following equations in the range 0◦
to 360◦
(a) sin−^1 (− 0 .4161)=x
(b) cot−^1 (2.4198)=θ (8) - Sketch the following curves labelling relevant
points:
(a)y=4 cos(θ+ 45 ◦)
(b)y=5 sin(2t− 60 ◦) (8) - The current in an alternating current circuit at
any timetseconds is given by:
i=120 sin(100πt+ 0 .274) amperes.
Determine
(a) the amplitude, periodic time, frequency and
phase angle (with reference to 120 sin 100πt)
(b) the value of current whent= 0
(c) the value of current whent=6ms
(d) the time when the current first reaches 80 A
Sketch one cycle of the oscillation. (19) - A complex voltage waveformv is comprised
of a 141.1 V rms fundamental voltage at a fre-
quency of 100 Hz, a 35% third harmonic com-
ponent leading the fundamental voltage at zero
time byπ/3 radians, and a 20% fifth harmonic
component lagging the fundamental at zero time
byπ/4 radians.
(a) Write down an expression to represent
voltagev
(b) Draw the complex voltage waveform using
harmonic synthesis over one cycle of the
fundamental waveform using scales of 12 cm
for the time for one cycle horizontally and
1cm=20 V vertically (15)- Prove the following identities:
(a)√[
1 −cos^2 θ
cos^2 θ]
=tanθ(b) cos(
3 π
2+φ)
=sinφ(c)sin^2 x
1 +cos 2x=^12 tan^2 x (9)- Solve the following trigonometric equations in
the range 0◦≤x≤ 360 ◦:
(a) 4 cosx+ 1 = 0
(b) 3.25 cosecx= 5. 25
(c) 5 sin^2 x+3 sinx= 4
(d) 2 sec^2 θ+5 tanθ= 3 (18) - Solve the equation 5 sin(θ−π/ 6 )=8 cosθfor
values 0≤θ≤ 2 π (8) - Express 5.3 cost− 7 .2 sintin the form
Rsin(t+α). Hence solve the equation
5 .3 cost− 7 .2 sint= 4 .5 in the range
0 ≤t≤ 2 π (12) - Determine
∫
2 cos 3tsintdt (3)