The Sine Ratio

Further Ideas

The programme deals only with the sine ratio and calculation of lengths and angles. Students are expected to have done some work within this topic prior to viewing, and many will already have met all three trigonometric ratios. The underlying concepts of similarity and ratios of corresponding sides in similar triangles can be explored using spreadsheets.

Groups or the whole class can begin by each drawing a right-angled triangle containing a given angle. No other conditions are set — so a set of similar but not identical triangles will be produced. Students then record the lengths of the sides as accurately as possible, using the labels ‘opposite’, ‘adjacent’ and ‘hypotenuse’ in relation to the given angle. The results are pooled and entered in columns on a spreadsheet (this could be done by the teacher prior to the next lesson if the group is large).

Students investigate the results of programming the spreadsheet to divide pairs of sides. This should lead them to recognise that no matter what size the triangle, the ratios of equivalent pairs of sides are equal. It is helpful if the columns are labelled with the calculation performed to generate them — for example, ‘opp/hyp’. Finding all six possible ratios here can generate later discussion of why only three are needed on calculators or in books of tables. The spreadsheet allows a large number of results to be processed and compared, showing the tendency in each column to a particular value. Any mistakes in measuring or recording should stand out, and students should appreciate that the degree of accuracy of their measurements limits that of the results.

An investigation such as this, where students generate the values from their own triangles, demonstrates that trigonometry is not a ‘mathematician’s trick’ but is founded in actual measurement and observation. The sine, cosine and tangent functions can be introduced and defined, and a calculator used to compare values in the spreadsheet with the stored values for the given angle. To introduce a discussion of how the values can be used to find one side given another side, you could say: ‘I’m going to draw a triangle, containing our angle, with a hypotenuse of x cm. Try to work out how long the other two sides will be.’ Students could comment on what they think would happen if you had chosen a different starting angle, and could then check their conjectures by drawing and measuring sets of triangles. The complementary angle to the one given could also be introduced, and the values in the spreadsheet used to develop the relations between trigonometric values for complementary angles.

The activity can be extended to triangles containing other angles. You could set students the task of creating a set of tables for the sin, cos and tan of 10º, 20º, 30º, and so on, by drawing triangles and also by using a spreadsheet. The results from different groups of students could be compared, and calculators used to check which values are closest to the true values. The aim is to give students a more concrete understanding of the trigonometric ratios: the contents of their table and the values given by their calculators are essentially the same, although the calculator is more accurate. The underlying concept of similarity should also be strengthened.

The context of fire-fighters using ladders could be used to explore more involved calculations. In situations where greater heights need to be reached, a turntable ladder mounted on an engine is often used. You could ask questions such as: ‘If the turntable ladder is extended to 30m and positioned at an angle of 70º, what height can the ladder can reach, given that it is mounted on the engine 2m above the ground?