btec hnc - analytical methods - trigonometric methods

Trigonometric MethodsAnalytical Methods for Engineers

By Brendan Burr

Brendan Burr BTEC Higher National Certificate in ElectronicsTrigonometric Methods

Table of Contents

TABLE OF CONTENTS - 2 -

TASK 1 - 6 -

1.1 A 4.2 m long ladder is placed against a perpendicular wall with its foot 60 cm from the wall. - 6 -

(a) Find how far up the wall the ladder reaches in feet. - 6 -Solution:- - 6 -Check:- - 6 -

(b) If the foot of the ladder is moved 20 cm towards the wall how far does the top of the ladder rise in inches ? - 7 -

Solution:- - 7 -Check:- - 7 -

1.2 From a point on horizontal ground a surveyor measures the angle of elevation of an aerial as 20 º. He moves 50 m closer to the aerial and measures the angle of elevation as 25 º. Determine the height of the aerial. - 8 -


1.3 A ship X sails at a steady speed of 50 km/hr in a direction of W 30 º N, i.e a bearing of 300 º from port. At the same time another ship Y leaves port at a steady speed of 40 km/hr in a direction of N 20 º E i.e a bearing of 20 º from port. Determine their distance apart after 5 Hrs. - 11 -

Solution:- - 11 -

1.4 - 12 -

(a) Determine the resultant of the 2 forces shown - 12 -Solution:- - 12 -

(b) Determine the angle it makes with the 50 N force. - 13 -Solution:- - 13 -

1.5 Solve triangle ABC given that angle C=90°, angle A=35° and AC=5 cm. - 14 -Solution:- - 14 -

2


TASK 2 - 15 -

2.1 (a) Convert the following Cartesian co-ordinates into Polar co-ordinates correct to 1 decimal place in degrees. (-2.2, 5.5) - 15 -


(b) Convert the following Polar co-ordinates into Cartesian co-ordinates correct to 3 decimal places. (6.4, 130 °) - 16 -


2.2 If the diameter of a circular wheel is 100 cm determine :- - 17 -

(a) The radius - 17 -Solution:- - 17 -

(b) The circumference - 17 -Solution:- - 17 -

2.3 A train is travelling at 108 km/hr and has wheels of diameter 80 cm. Determine the angular velocity of the wheels in:- - 18 -

(i) Radians/sec - 18 -Solution:- - 18 -

(ii) Revs/min. - 18 -Solution:- - 18 -

(b) If the speed remains constant for 2.7 km determine the no. of revolutions made by a wheel assuming no slipping occurs. - 19 -

Solution:- - 19 -

2.4 The voltage in an alternating circuit at any time t seconds is given by: volts Determine :- - 20 -

(a) Amplitude - 20 -Solution:- - 20 -

(b) Periodic time - 20 -Solution:- - 20 -

(c) Frequency - 20 -Solution:- - 20 -

(d) The voltage when t = 0 s - 21 -Solution:- - 21 -

3


(e) The voltage when t = 5 ms - 21 -Solution:- - 21 -Check:- - 21 -

2.5 A complex voltage waveform v is comprised of a 141.4 v rms fundamental voltage at a frequency of 200 Hz, a 40 % third harmonic component leading the

fundamental voltage at zero time by radians and a 20 % fifth harmonic

component lagging the fundamental by radians. - 22 -

a) Using harmonic synthesis determine the expression that represents the voltage waveform v. - 22 -

Solution:- - 22 -

b) Using Graphmatica S/W obtain a graph of the complex voltage waveform representing the expression for voltage over one cycle. - 24 -

Solution:- - 24 -

TASK 3 - 25 -

3.1 Prove the following trigonometric identity : - 25 -

- 25 -Solution:- - 25 -Check:- - 25 -

3.2 Solve the following trigonometric equation for values of t from 0 ° to 360 °. - 26 -


3.3 Solve the trigonometric equation for values of θ

from 0 to 2π rads. - 28 -Solution:- - 28 -

3.4 Express in the form - 30 -Solution:- - 30 -Check:- - 31 -

3.5 Hence solve the equation in the range - 32 -

. - 32 -Solution:- - 32 -Check:- - 33 -

4


EVALUATION - 34 -

CONCLUSION - 34 -

BIBLIOGRAPHY - 35 -

Books - 35 -

Catalogues - 35 -

Websites - 35 -

5


Task 1

1.1 A 4.2 m long ladder is placed against a perpendicular wall with its foot 60 cm from the wall.

(a) Find how far up the wall the ladder reaches in feet.

Solution:-

Check:-

Convert meters to feet.

6


(b) If the foot of the ladder is moved 20 cm towards the wall how far does the top of the ladder rise in inches ?

Solution:-

The Ladder rose up the wall by inches.

Check:-

Convert meters to feet.

Convert feet to inches.

7


1.2 From a point on horizontal ground a surveyor measures the angle of elevation of an aerial as 20 º. He moves 50 m closer to the aerial and measures the angle of elevation as 25 º. Determine the height of the aerial.

Solution:-

8


9


Check:-

So:

10


1.3 A ship X sails at a steady speed of 50 km/hr in a direction of W 30 º N, i.e a bearing of 300 º from port. At the same time another ship Y leaves port at a steady speed of 40 km/hr in a direction of N 20 º E i.e a bearing of 20 º from port. Determine their distance apart after 5 Hrs.

Solution:-

11


1.4

(a) Determine the resultant of the 2 forces shown

Solution:-

12


(b) Determine the angle it makes with the 50 N force.

Solution:-

13


1.5 Solve triangle ABC given that angle C=90°, angle A=35° and AC=5 cm.

Solution:-

14


Task 2

2.1 (a) Convert the following Cartesian co-ordinates into Polar co-ordinates correct to 1 decimal place in degrees. (-2.2, 5.5)

Solution:-

By Pythagoras:

UNITS

Check:-

Using the Calculator:

15


(b) Convert the following Polar co-ordinates into Cartesian co-ordinates correct to 3 decimal places. (6.4, 130 °)

Solution:-

Check:-

Using the Calculator:

16


2.2 If the diameter of a circular wheel is 100 cm determine :-

(a) The radius

Solution:-

(b) The circumference

Solution:-

17


2.3 A train is travelling at 108 km/hr and has wheels of diameter 80 cm.Determine the angular velocity of the wheels in:-

(i) Radians/sec

Solution:-

(ii) Revs/min.

Solution:-

18


(b) If the speed remains constant for 2.7 km determine the no. of revolutions made by a wheel assuming no slipping occurs.

Solution:-

19


2.4 The voltage in an alternating circuit at any time t seconds is given by: volts

Determine :-

(a) Amplitude

Solution:-

Amplitude = 100V

(b) Periodic time

Solution:-

(c) Frequency

Solution:-

20


(d) The voltage when t = 0 s

Solution:-

(e) The voltage when t = 5 ms

Solution:-

Check:-

21


2.5 A complex voltage waveform v is comprised of a 141.4 v rms fundamental voltage at a frequency of 200 Hz, a 40 % third harmonic

component leading the fundamental voltage at zero time by radians

and a 20 % fifth harmonic component lagging the fundamental by

radians.

a) Using harmonic synthesis determine the expression that represents the voltage waveform v.

Solution:-

Fundamental Frequency

(Grid Range)

Harmonic

22


Harmonic

23


b) Using Graphmatica S/W obtain a graph of the complex voltage waveform representing the expression for voltage over one cycle.

Solution:-

24


Task 3

3.1 Prove the following trigonometric identity :

Solution:-

Check:-

Solve Triangle a=5, b=4 and c=3

25


3.2 Solve the following trigonometric equation for values of t from 0 ° to 360 °.

Solution:-

Use Formula Method to find values of x:

26


Check:-

27


3.3 Solve the trigonometric equation for values of

θ from 0 to 2π rads.

Solution:-

LEFT HAND SIDE:

BOTH SIDES:

28


29


3.4 Express in the form

Solution:-

OR:

Where = 0.9566368995 OR -2.184955754

30


Check:-

31


3.5 Hence solve the equation in the range

.

Solution:-

32


Check:-

33


EvaluationIn Task 1, I was faced with calculating in imperial, which is something I very rarely have to do. Due to my lack of knowledge with this form of measurement I calculated it fully in both ways (metric and imperial) to make sure I got the correct answer at the end.I was also asked to calculate the height of an aerial with minimal information. This meant having to create three triangles and solving all three to get the answer.I found that I was able to test my competence with using the Sine and Cosine Rules as well as basic Pythagoras Theorem work.In Task 2, I had to represent the direct relationship between Cartesian co-ordinates and Polar co-ordinates. This was relatively easy to pick up due to previous work completed on Cartesian and Polar.There was also a requirement to understand the calculation and relationship between Revolutions of a wheel and Distance, which I was able to understand.Linking with work we have previously done in the course, we were asked to determine the various characteristics of a voltage waveform, such as Amplitude, Periodic Time, Frequency and the Voltage at two different instances. I could then enter the voltage expression into Graphmatica and evaluate points of y dependant on the value of x, which was either 0S or 5mS, to allow me to check my workings.Harmonic waveforms are something I can relate slightly to work, as testing on audio systems on the aircraft make irregular sinusoidal waveforms which are presented on the test equipment. Working out the expression that goes with this is potentially beneficial to this work.For Task 3, I had to prove trigonometric identities and equations. The problem I found was when it came to checking the answer for 3.2. This was because I couldn’t represent the final evaluated x value in Degrees, it would only come up in Radians. I was forced to leave it in the end and presume that because I had converted it from Radians to Degrees, the check would count as correct.Expressing trigonometric equations in a different form involved following and understanding examples, as a minor error could result in a completely different answer when it came to question 3.4 and 3.5. I performed a check at the end of each, to ensure that the correct answer had been achieved.

ConclusionI found that I spent a lot of time in the presentation as I have done with all my previous assignments. This helps to make the answers clear and separates them from the workings out. I will continue to do this in future assignments as it helps me when checking the work at the end as well.I found that I couldn’t check all of my answers as some only had one method of working out. I will however continue checking as many answers as I can as this ensures correctness and punctuation in the work completed.

34


Bibliography

Through guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:

Books

BTEC National Engineering (Mike Tooley & Lloyd Dingle) ISBN: 978-0-7506-8521-4Success in Electronics (Tom Duncan & John Murray)ISBN: 0-7195-4015-1Higher Engineering Mathematics (John Bird) ISBN: 0-7506-8152-7

Catalogues

N/A

Websites

N/A

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