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)