Revision Test 5
This Revision Test covers the material contained in Chapters 14 to 17. The marks for each question are shown in
brackets at the end of each question.
1. Solve the following equations in the range 0 ◦
to 360 ◦ .
(a) sin −1 (−0.4161) = x
(b) cot −1 (2.4198) = θ
(8)
2. Sketch the following curves labelling relevant
points:
(a) y = 4 cos(θ + 45 ◦ )
(b) y = 5 sin(2t − 60 ◦ )
(8)
3. The current in an alternating current circuit at
any time t seconds 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 when t = 0
(c) the value of current when t = 6 ms
(d) the time when the current first reaches 80 A
Sketch one cycle of the oscillation.
(19)
4. A complex voltage waveform v is comprised
of a 141.4 V rms fundamental voltage at a frequency of 100 Hz, a 35% third harmonic component leading the fundamental voltage at zero
time by π/3 radians, and a 20% fifth harmonic
component lagging the fundamental at zero time
by π/4 radians.
(a) Write down an expression to represent
voltage v.
(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
1 cm = 20 V vertically.
(15)
5. Prove the following identities:
(a)
1 − cos 2 θ
cos 2 θ
= tan θ
(b) cos
3π
2
+ φ
= sin φ
(c)
sin
2 x
1 + cos 2x
=
1
2 tan 2 x
(9)
6. Solve the following trigonometric equations in the
range 0 ◦ ≤ x ≤ 360 ◦ :
(a) 4 cos x + 1 = 0
(b) 3.25 cosec x = 5.25
(c) 5 sin 2 x + 3 sin x = 4
(d) 2 sec 2 θ + 5 tanθ = 3
(18)
7. Solve the equation 5 sin(θ − π/6) = 8 cosθ for
values 0 ≤ θ ≤ 2π.
( 8 )
8. Express 5.3 cos t − 7.2 sint in the form
R sin(t + α). Hence solve the equation
5.3 cos t − 7.2 sint = 4.5 in the range
0 ≤ t ≤ 2π.
(12)
9. Determine
2 cos3t sin t dt .
( 3 )
This Revision Test covers the material contained in Chapters 14 to 17. The marks for each question are shown in
brackets at the end of each question.
1. Solve the following equations in the range 0 ◦
to 360 ◦ .
(a) sin −1 (−0.4161) = x
(b) cot −1 (2.4198) = θ
(8)
2. Sketch the following curves labelling relevant
points:
(a) y = 4 cos(θ + 45 ◦ )
(b) y = 5 sin(2t − 60 ◦ )
(8)
3. The current in an alternating current circuit at
any time t seconds 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 when t = 0
(c) the value of current when t = 6 ms
(d) the time when the current first reaches 80 A
Sketch one cycle of the oscillation.
(19)
4. A complex voltage waveform v is comprised
of a 141.4 V rms fundamental voltage at a frequency of 100 Hz, a 35% third harmonic component leading the fundamental voltage at zero
time by π/3 radians, and a 20% fifth harmonic
component lagging the fundamental at zero time
by π/4 radians.
(a) Write down an expression to represent
voltage v.
(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
1 cm = 20 V vertically.
(15)
5. Prove the following identities:
(a)
1 − cos 2 θ
cos 2 θ
= tan θ
(b) cos
3π
2
+ φ
= sin φ
(c)
sin
2 x
1 + cos 2x
=
1
2 tan 2 x
(9)
6. Solve the following trigonometric equations in the
range 0 ◦ ≤ x ≤ 360 ◦ :
(a) 4 cos x + 1 = 0
(b) 3.25 cosec x = 5.25
(c) 5 sin 2 x + 3 sin x = 4
(d) 2 sec 2 θ + 5 tanθ = 3
(18)
7. Solve the equation 5 sin(θ − π/6) = 8 cosθ for
values 0 ≤ θ ≤ 2π.
( 8 )
8. Express 5.3 cos t − 7.2 sint in the form
R sin(t + α). Hence solve the equation
5.3 cos t − 7.2 sint = 4.5 in the range
0 ≤ t ≤ 2π.
(12)
9. Determine
2 cos3t sin t dt .
( 3 )