In this lab we are going to practice the trigonometric skills we'll need in physics.  First, we'll use trigonometry to calculate the heights of several tall objects like trees and flagpoles.  While we're at it, we'll do some statistical analysis of our results.  Then we'll use the Law of Cosines and Law of Sines to deal with triangles that are not right triangles.  And as always, error analysis will rear its ugly head.

Part A: Finding Height Using Right Triangles

Surveyors may be faced with the challenge of finding a distance that they can't measure directly, such as the height of a tall object or the distance to a building on the far side of a river or chasm.  By taking some simple measurements, they can define a right triangle and use trigonometry to calculate the length of the desired side.

Consider the person looking up at a tall tree drawn below:

q

b

It's easy enough to measure the baseline distance b that he's standing from the base of the tree.  By using a device called an inclinometer, he can measure the angle between the base and the hypotenuse.  Since the triangle is right, that's all we need to find any side we want.  In this case, we can use the tangent function to find the height of the tree.

One complicating factor is that the base of the triangle is at the student's eye level, not on the ground.  So when we calculate the height, it's really the height of the tree above eye level.  We need to measure the distance from the ground to the student's eye level and add that to our calculated height.  The final equation is

	h = b tan q + d

Make sure all distances are measured using the same units, of course.

Methodology

1.	Split up into at least four groups.  Make sure each group has an inclinometer and something to measure distance: a tape measure or some meter sticks.  Once groups are selected, we'll go outside and enjoy the sunshine.

2.	Each group should select something tall to measure: a tree, a flagpole, a streetlight, etc.  Make sure that you can actually get to the object and that there is at least one area of level ground adjacent to it.

3.	One member of the group should stand some distance from the object with the inclinometer.  The rest of the group can measure the distance from his feet to the base of the object.  Also, someone should measure the distance from the ground to the student's eyes.  Don't forget to include uncertainty on each measurement.

4.	Using the inclinometer, measure the angle of inclination from your eyes to the top of the object.  As always, include uncertainty on the measurement.

5.	When you're finished, switch with another group and measure their object.  Don't use their baseline distance, select your own.  Cycle through all four objects and get measurements for each.

6.	Share your data so that everyone has all four group's data for all four objects.  

Analysis

Use a spreadsheet to find the measured values for each object's height.  Find the average value of each and the standard deviation for the results.  Make sure you round each cell entry to proper significant figures.  Show sample calculations, including error analysis, for one set of data.

Part B: Law of Sines and Cosines

When a triangle isn't right, we can't use Pythagorean Theorem or simple definitions like opposite over hypotenuse.  To solve these triangles, we need to use the Law of Sines

	sin A              sin B             sin C
	-------     =     -------     =     -------
	   a                    b                    c

and the Law of Cosines

	c2 = a2 + b2 - 2ab cos C

where A, B, and C are angles of the triangle and a, b, and c are the sides opposite those angles.

Methodology

1.	Find a triangle that isn't right somewhere in the school or on the school grounds.  Everyone should have his own unique triangle.  If you can't find one, make one yourself.

2.	Sketch the triangle in your lab notebook.  Label the angles A, B, and C.  Label the side opposite A as side a; repeat for sides b and c.

3.	Measure the lengths of sides a and b and record the lengths with proper uncertainty.  Measure the size of angle C.  These are your known quantities for the triangle.

4.	We're going to calculate the remaining angles, A and B, and the remaining side, c.  You will, however, want to know the correct values for them so you can see how accurate your results are.  Measure side c and angles A and B with proper uncertainty, record them in your lab notebook, and forget that you know them until the end of the lab.

Analysis

Use the Law of Sines and the Law of Cosines to find the unknown side and angles of the triangle.  Show all your work and include error analysis.  Determine whether the correct value for each one lies within the error range of your calculated values.

Questions

1.	Did a long or a short baseline give better results for the height of the tall object?  Explain why based on the equation used.

2.	Imagine a group accidentally used a yardstick instead of a meter stick to measure their baseline.  Would their calculated height values be too large or too small?  Explain.

3.	Does your standard deviation relate more to the accuracy or the precision of your results?  Explain.

4.	In part B, did you choose your uncertainty values wisely?  Justify your answer.

5.	What had greater uncertainty, the lengths or the angles?  Why is that?