gac004 project 2 ae2 guide -...

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GAC004 AE2 Project 2: This exercise has the purpose of getting students working with the Sine and Cosine rules of Trigonometry, with the additional side benefit of subtly informing them of the imprecision involved in working with experimental data. Teachers should note that the paper protractor provided to be used in this exercise will be reasonably imprecise in estimating the actual angles, and this could result in a good degree of error in the results. In my opinion, students should not be punished for this in marking, however I think we should expect them to comment briefly on the errors they found in the discussion part of their report. Only two pieces of graphical evidence are really necessary for the final report – a picture, drawing, or photograph, of the two triangles laid out on the ground. Experimental equipment and measurements only need descriptions. We know what a measuring tape is, and we assume the student knows how to use one – we don’t need photographic evidence of this! As an example of the sort of measurements to expect, here are two triangles I laid out and measured at a local park here in Sydney:

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Page 1: GAC004 Project 2 AE2 guide - itapglobal.comitapglobal.com/.../GAC004/GAC004_Project_2_AE2_Guide.pdf · GAC004 AE2 Project 2: ... students may make the mistake of simply writing down

GAC004 AE2 Project 2:

This exercise has the purpose of getting students working with the Sine and Cosine rules of Trigonometry, with the additional side benefit of subtly informing them of the imprecision involved in working with experimental data. Teachers should note that the paper protractor provided to be used in this exercise will be reasonably imprecise in estimating the actual angles, and this could result in a good degree of error in the results. In my opinion, students should not be punished for this in marking, however I think we should expect them to comment briefly on the errors they found in the discussion part of their report.

Only two pieces of graphical evidence are really necessary for the final report – a picture, drawing, or photograph, of the two triangles laid out on the ground. Experimental equipment and measurements only need descriptions. We know what a measuring tape is, and we assume the student knows how to use one – we don’t need photographic evidence of this!

As an example of the sort of measurements to expect, here are two triangles I laid out and measured at a local park here in Sydney:

Page 2: GAC004 Project 2 AE2 guide - itapglobal.comitapglobal.com/.../GAC004/GAC004_Project_2_AE2_Guide.pdf · GAC004 AE2 Project 2: ... students may make the mistake of simply writing down

1. An Obtuse Triangle.

Photograph as shown – the tape measure has been extended exactly 1 metre to give an idea of scale in the picture (I would recommend this to students).

As per the instructions, I measured:

Length AC = 444 cm, Length AB = 275 cm, ∠𝐵𝐴𝐶 = 27°

Therefore, by the Cosine rule, we can find the remaining side length:

𝐵𝐶! = 444! + 275! − 2 ∙ 444 ∙ 275 ∙ cos 27°

∴ 𝐵𝐶 ≅ 234.9 cm

The measured length of BC was 233 cm, so the calculated value is fairly close, given the uncertainties in the measurements, particularly that of the angle BAC.

The obtuse angle can now be found by the Sine rule:

!"#∠!"#!!!

= !"# !"!"#.!

∴ ∠𝐴𝐵𝐶 ≅ 180°− 59.1° = 120.9°

Note that these two calculations are ordered the reverse of how they are presented in the instructions. In my opinion this is a much more logical way to do them.

Important! Note that to get the obtuse angle from the sine rule, you need to take the result of the sin-1 away from 180° (sin-1 will only give an answer between 0° and 90°). Many

A  

B  

C  

Page 3: GAC004 Project 2 AE2 guide - itapglobal.comitapglobal.com/.../GAC004/GAC004_Project_2_AE2_Guide.pdf · GAC004 AE2 Project 2: ... students may make the mistake of simply writing down

students may make the mistake of simply writing down the result of the calculation without thinking about it.

I actually measured the obtuse angle in the triangle to be 121°, so my measurements here turned out to be quite accurate (unexpectedly – I was anticipating the results to be off by quite a bit more).

We can now calculate the area of the triangle:

𝐴 =12 ∙ 444 ∙ 275 ∙ sin 27° ≅ 27  716  cm! = 2.77  m!

Or alternatively, we measure the vertical height of the triangle from side AC, which is 125 cm. From this we obtain an Area:

𝐴 =12 ∙ 444 ∙ 125 ≅ 27  750  cm! = 2.78  m!

Again, a small error due to measurement inaccuracy is to be expected.

Page 4: GAC004 Project 2 AE2 guide - itapglobal.comitapglobal.com/.../GAC004/GAC004_Project_2_AE2_Guide.pdf · GAC004 AE2 Project 2: ... students may make the mistake of simply writing down

2. An Acute Triangle.

Photograph as shown (again the tape is extended one metre to indicate scale):

The instructions are a little unclear here, as they say we should calculate the three angles, but does not specify what measurements we should use to do so. At least three measurements are needed to solve all the dimensions of a triangle. It makes sense to choose two sides and an angle, as in the first example, so that’s what we’ll do here:

Length AB = 375 cm, Length AC = 341 cm, ∠𝐵𝐴𝐶 = 39°

𝐵𝐶! = 375! + 341! − 2 ∙ 375 ∙ 341 ∙ cos 39°

∴ 𝐵𝐶 ≅ 241.1 cm

!"#∠!"#!"#

= !"# !"!"#.!

∴ ∠𝐴𝐵𝐶 ≅ 62.9°

And the third angle can be easily obtained from (alternatively use the Sine rule again):

∴ ∠𝐴𝐶𝐵 ≅ 180°− 39°− 62.9° = 78.1°

Actual measurements of the other two angles yielded ∠𝐴𝐵𝐶 = 61° and ∠𝐴𝐶𝐵 = 82°, so we’re a couple of degrees out this time. Still close enough that we can believe that the Sine and Cosine rules work in practice.

A  

B  

C