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Exact Trigonometric Ratios

This morning I read this interesting little post from Andy Lyons (@mrlyonsmaths) which looked at teaching the exact Trigonometric Ratios for certain given angles (namely 0, 30, 60, 90 and 180 degrees). The post gave a nice little info graphic linked to the unit circle to show what was going on and then focused on methods yo remember the ratios.

While reading it I thought about how I introduce these exact Trigonometric Ratios. I first like to know that my students have a thorough and in depth understanding of right angled triangles and the trigonometry involved with them (including Pythagoras’s Theorem). I feel this is imperative to learning mathematics, the Triangle is an extremely important shape in mathematics and to fully understand triangles you must first fully understand the right angled triangle. The rest follows from that.

Once these are understood then you can move on to the trigonometric graphs, showing how these can be generated from right angled triangles within the unit circle, as shown in the info graphic on Andy’s post. Once the graphs are understood then the coordinates f the x and y intercepts and the turning points give us nice exact values for angles of 0, 90 and 180 degrees. This leaves us with 30, 60 and 45 to worry about.

At this point I introduced 2 special right angled triangles. First up is the right angled isosceles triangle with unit lengths of the short sides. This obviously gives us a right angled triangle that has two 45 degree angles (as the angle sum of a triangle is 180) and a hypotenuse of rt2 (via Pythagoras’s Theorem).

image

Using our definitions of trigonometric ratios (ie sin x =opp/hyp, cos x = adj/hyp and tan x = opp/adj) we can clearly see that tan 45 = 1 and that sin 45 = cos 45 = 1/rt2. This aids the understanding more than just giving the values and allows students a method of working these values out easily if stuck.

The second triangle is an equilateral triangle of side length 2 cut in half. This gives us a right angled triangle with hypotenuse 2, short side lengths 1 and rt3 (again obtained through Pythagoras’s Theorem) and angles 30, 60 and 90.

image

Again we can use our definitions of trigonometric ratios to conclude that sin 30 = cos 60 = 1/2, sin 60 = cos 30 = rt3/2, tan 30 = 1/rt3 and tan 60 = rt3.

This is again good for deeper understanding and for seeing why sin x = cos 90 – x, and cos x = sin 90 -x. This can lead to a nice discussion around complementary angles and that the word cosine means “sine of the complementary angle”. This triangle is also a good demonstration that tan x = cot 90 – x, when you come to higher level trig.

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  1. Ronald
    November 8, 2015 at 1:10 pm

    As a tutor, I have to pick up a lot of misconceptions that result from this teaching method.

    Students often focus on methods applied to the special triangles (equilateral, isosceles), and then get very stuck and confused trying to apply these methods to other angles. Occasionally they think that these angles are ‘special’ in that a regular calculator application won’t work here. Very often they take these special triangles as examples from which to learn a (not applicable) method.

    Students learning trigonometry don’t have the experience to understand that special triangles are just that – special cases only.

    You say that a thorough and in depth understanding is required, and I say – while these methods are useful for depth and flavour later on, they are not thorough enough for beginners to grasp the limitations.

    • November 8, 2015 at 2:02 pm

      Indeed, that’s why I wouldn’t introduce them until the thorough and in depth knowledge has been achieved.

  2. November 8, 2015 at 2:18 pm

    My Russia/Ukrainian students are often surprised that we use calculators for trigonometry in C2 etc. They are used to working on problems that always boil down to exact values and calculations that can be done on paper. It’s funny that they see Trig as an abstract mathematical process and don’t at first appreciate why we might want a triangle with an angle of say 35 degrees!

  3. Mark Wilson
    November 8, 2015 at 10:37 pm

    Teaching is surely about communicating the ideas and making formative assessments to check understanding. I have taught Trig to beginners in a number of different ways. For some students it takes forever, whilst with others they grasp it straight away, irrespective of the method of introduction. Teaching is an art.

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