Trigonometry

4.10 TRIGONOMETRY

TRIGONOMETRY LESSON IDEA 1

TITLE: INTRODUCING THE TAN RATIO
TOPIC: TRIGONOMETRY
AIM: To give students an opportunity to find the angle of elevation of the sun from experimental field-work.

RESOURCES:
Poles of various lengths (e.g. sweeping brush, metre stick), measuring tape, graph paper, protractor, centimetre ruler, pencil, eraser, a sunny day and safety instructions about not looking at the sun!

METHOD:
Outside instructions

1. Students work in twos. One person holds the pole vertically on level ground so that its shadow can be clearly seen.
2. Record the date and time. Measure and record the length of the pole.
3. Measure and record the length of the shadow.
4. Draw a rough diagram.

Back in the classroom

1. Decide on a scale to use.
2. Draw an accurate diagram on graph paper.
3. Measure the angle of elevation of the sun (A) using a protractor.
4. Compare your answer with that of others in the class.

Make up a table showing times and angles.
The results should yield (approximately) the same angle of elevation.
Fill in the table below. (The calculator can be used to evaluate tanA so that results in columns 3 and 5 can be compared.)

Students can experiment by drawing triangles with the same angle of elevation but various lengths on the horizontal and vertical, and then completing a similar table.

CLASSROOM MANAGEMENT IMPLICATIONS:
The initial field-work can be done in pairs but then students can be encouraged to compare results with their peers.

TRIGONOMETRY LESSON IDEA 2

TITLE: TRIGONOMETRY AND THE USE OF THE EYE!
TOPIC: TRIGONOMETRY
AIM: To show students how to calculate sides of a triangle using alternative definitions.

RESOURCES:
None.

METHOD:

1. Start with the standard trigonometric ratios (based on a right-angled triangle):
   Sine of angle = Opposite/Hypotenuse
   Cosine of angle = Adjacent/Hypotenuse
   Tangent of angle = Opposite/Adjacent
2. Derive the following from the standard formulae:
   Hypotenuse × Sine of angle = Opposite
   Hypotenuse × Cosine of angle = Adjacent
   Adjacent × Tangent of angle = Opposite
3. This approach enables students to determine the other two sides of the triangle when provided with the hypotenuse b and the base angle A.
4. Frequently, students are asked to determine the height of a building. This involves standing back from the building and measuring
   i. the angle of elevation A
   ii. the horizontal distance c to the base of the building.
   Using a calculator, the students can key in the data to yield an immediate result:
   height = c tan A

CLASSROOM MANAGEMENT IMPLICATIONS:
None
Additional, follow-up work is given below, showing how the Sine Rule and the formula for the area of a triangle might be derived (the proofs are not examinable).

NOTE:

1. There is little doubt but that students have difficulty with applying trigonometric ideas. Much of the problem is with handling orientation in space. An eye is drawn on the blackboard to indicate the side from which the diagram is viewed. In the example given below the task is to find the perpendicular height.
2. Equally, if the triangle is viewed from the left then the area of the triangle can be given by:
   Area= (1/2)base × perpendicular height
   = (1/2)c × b sin A
   = (1/2)bc sin A
   Viewing from the right produces:
   Area= (1/2)base × perpendicular height
   = (1/2)c × a sin B
   = (1/2)ac sin B

TRIGONOMETRY LESSON IDEA 3

TITLE: THE CLINOMETER AND TRIGONOMETRY
TOPIC: TRIGONOMETRY
AIM:

1. To give students an opportunity to make a clinometer and to measure elevation and declination.
2. To allow students the opportunity to calculate the heights of structures with the help of the clinometer in experimental fieldwork.
3. To give students an opportunity to experience mathematical modelling.

RESOURCES:
Stiff cardboard (280 gsm or heavier), paper, pencil, black thread, paper fasteners, plasticine and scissors.

METHOD:

1. The model clinometer shown in Figure 1 should be made from stiff card (280gsm or heavier). The viewingtube can be made from a drinking straw or by rolling writing paper around a pencil. The "plumbline" should be made from black thread and attached to the centre of the semi-circle by a paper fastener and finally the weight might be plasticine. The degree markings should be made using a protractor and set at 5º spacings; in use, smaller measurements might have to be estimated.
   One of the many interesting problems for the students is whether to mark 0º to180º around the semicircle or to mark 0º to 90º on two halves, so eliminating the subtraction of readings in order to calculate angle a.
2. Figure 2 shows, schematically, the measurements students are expected to make: the distance D from the structure, h1 (their eye-level), and finally a, the angle of elevation. It is surprising how many problems arise with simple measuring. Many students will never have used metre sticks and can often have trouble marrying metres, centimetres and millimetres into the correct written decimal form.
3. Students can then be introduced to the idea of "mathematical modelling" of the situation when they use the formula to determine H, the height of the structure:
   H = h1 + h2
   The value of h2 can be found graphically by scaling the whole exercise on graph paper.
4. The more able students can perhaps find the value of h2 by using:
   tana = (h2)/(D)
   The latter formulation should be tried eventually by all students, since it gives them a concrete example of what "tan" is in the case of right-angled triangles and hence how it can be used to calculate the lengths of the sides of a triangle.
5. For the more able mathematicians Figure 3 shows how, if the measurement D to the base of the structure cannot be measured, then determining two angles a and ß from two measured points (distance E apart) will allow use of the sine rule in triangle abd in order to find bd, the hypotenuse of triangle bcd. This, in turn, allows the use of tanß to calculate h2 in triangle dbc.

NOTE:
The variety of mathematical words and ideas that occur when undertaking the making and using of the clinometer is immense. Many skills are employed by students: psychomotor, organisational and graphical skills; skills of measurement, scaling diagrams, recording values, and estimating accuracy. All these are allied to the algebraic skills of manipulating linear and trigonometrical expressions and formulae.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work in small groups of two or three for the fieldwork.