Grade: /22
Laboratory 2
Application of Trigonometry in Engineering
2.1 Laboratory Objective
The objective of this laboratory is to learn basic trigonometric functions, conversion from
rectangular to polar form, and vice-versa.
2.2 Educational Objectives
After performing this experiment, students should be able to:
1. Understand the basic trigonometric functions.
2. Understand the concept of a unit circle and four quadrants.
3. Understand the concept of a reference angle.
4. Be able to perform the polar to rectangular and rectangular to polar coordinate
conversion.
5. Prove a few of the basic trigonometric identities.
2.3 Background
Trigonometry is a tool that mathematically forms geometrical relationships. The understanding
and application of these relationships are vital for all engineering disciplines. Relevant
applications include automotive, aerospace, robotics, and building design. This lab will outline a
few common, but useful, trigonometric relationships.
2.3.1 Reference Angle
A reference angle is an acute angle (less than ) that may be used to compute the trigonometric
functions of the corresponding obtuse angle (greater than ). Figure 2.1 shows the reference
angle ϕ with respect to the angle θ. The reference angle is calculated using the formulas shown in
the captions of each corresponding subfigure of Figure 2.1
2.3.2 Law of Cosines
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The Law of cosines is a method that helps to solve triangles. Equations 2.1 relate the sides and
interior angles of Figure 2.2.
(2.1)
Figure 2.1: Law of Cosines Triangle
2.3.3 Law of Sines
The Law of Sines is another method that helps to solve triangles. Using the triangle of Figure
2.1, Equation 2.2 relates the sides to the interior angles.
(2.2)
2.4 Procedure
Follow the steps outlined below after the Lab Teaching Assistant has explained how to use the
laboratory equipment.
2.4.1 One Link Robot
1. Using the boards in the lab, fill in Table 2.1. Pay close attention to the sign of your
answer for all values.
NOTE: To convert a value in degrees to radians, the multiplying factor is π/180.
2. Use equation 2.3 to find the calculated x and y values.
(2.3)
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3. Using the boards in the lab, fill in Table 2.2.
4. Use Equations 2.4 to find the Calculated θ and l.
(2.4)
2.4.2 Trigonometric Identities
An identity is a trigonometric relationship that is true for all permissible values of the variable(s).
Many times, trigonometric identities are used to simplify more complex problems.
1. Using MATLAB, fill in Tables 2.3 and 2.4.
a. The first column of Table 2.3 comes from 5
th
column of Table 2.2.
b. Define this column as a vector in MATLAB and perform element by element
calculations on it to get the other columns.
NOTE: All calculations should be done with MATLAB. No calculator use!
2.4.3 Two Link Robot
1. Using the boards in the lab, fill in the Measured Values of Table 2.5.
NOTE: l
1
= l
2
= 50 mm
2. Write a MATLAB code to calculate x and y by adding the components of each link.
Recall the following equations from class.
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2.4.4 Solve a Triangle Using Law of Cosines and Law of Sines
In some cases, the laws of sines and cosines must both be used to solve a triangle. Figure 2.3 is one such
case where the lengths l
1
and l
2
along with the final ending point P of the two links are known and the
values are not. Both laws are needed to solve this triangle.
1. The radius r is found by:
2. Using the Law of Cosines, is found by the following equation:
(2.6)
3. Using the Law of Sines, is found by the following equation:
(2.7)
4. is now found by using the equations:
(2.8)
Figure 1.3: General Two Link Robot
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2.5 Lab Requirements
1. Write an abstract for this lab and submit it to the Lab 2 folder in your lab section’s abstract
folder found in the Pilot Dropbox. (Required to pass course.)
2. Complete Tables 2.1 – 2.6. (2 points each)
Table 2.1 Polar to Rectangular Conversion
Angle
θ°
Measured
x (mm)
Measured
y (mm)
Vector
Form
xî + yĵ
l
(mm)
Reference
Angle (°)
Reference
Angle
(rad)
Calculated
x (mm)
Calculated
y (mm)
30
100
45
100
90
100
135
100
180
100
225
100
270
100
Table 2.2 Rectangular to Polar Conversion
(x,y)
Measured
θ(°)
Reference
Angle (°)
Reference
Angle
(rad)
Calculated
θ(°)
Calculated
l (mm)
Polar Form
(85,50)
(70,70)
(0,100)
(-70,70)
(-100,0)
(-70,-70)
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Table 2.3
Calculated θ(°)
sin(θ)
cos(θ)
tan(θ)
sec(θ)
Table 2.4
Table 2.5
Measured Values
Calculated Values
X
Y
0
0
0
90
30
45
30
60
180
0
270
30
360
90
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Table 2.6 Application of Sine and Cosine Laws
P(x,y) mm
(55, 75)
(75, 60)
(15, 63)
(32, 14)
(71, 70)
3. Publish your MATLAB for Tables 2.3 through 2.6. (2 points each)
4. Answer the following questions.
a) Based on your results for Tables 2.3 and 2.4, write down the three trigonometric
identities that were verified. (2 points)
