There are three definitions of the trigonometric functions. One is based on the ratios of the opposite, adjacent and hypotenuse sides of a right-angled triangle. In the unit circle definition (described below), the sine (sinq) is the y co-ordinate of the end of a rotating rod and the cosine (cosq) is the x co-ordinate. Both depend on the size of the angle through which the rod turns, thus allowing for angles greater than 90o.
There is also a third, more rigorous, definition that uses differential equations.
The unit circle approach is very important as it helps us understand the graphs of the functions and hence the transformations of the curves.
I wonder how many students have asked a similar question, and been told only that tanq = sinqcosq and that this is the gradient of the line.
A unit circle is a circle with a radius of one unit, which we define. So if you draw a circle with a radius of 10cm, the scale would be 10cm to 1 unit.
Imagine that a rod, 1 unit long, is secured at the origin of the circle (0,0) and can rotate freely around it. Attached to its end is a piece of string with a weight on it, which meets the x-axis at right angles so that the length alters with the movement of the rod.
The rod acts as the hypotenuse of the right-angled triangles thus formed.
(The hypotenuse of a right-angled triangle is always the side opposite the right angle.) In this case, the hypotenuse AB is 1 unit length. The length of the side adjacent to angle q, adjusted for scale, gives us cosq. If the angle is 600, the length of cos600 is 0.5 (5cm for a 10cm rod); the opposite side gives us sin600, which is approximately 0.867 (8.67cm). These can be checked on a calculator by keying in cos600 and sin600.
The diagram shows the length of sinq in blue and of cosq in red, so we can easily see how these change as the rod moves round the circle creating different angles.
The unit can be one metre, one centimetre, one mile - whatever we choose - which leads on to this being applicable to real-life problems.
We can create the tangent length by drawing a tangent (a line that just touches a curve) with the point of tangency at Y, where q = 0, then extending the line AB to meet the tangent at X. Measuring XY gives us the value for tanq.
Triangle AXY is an enlargement of triangle ABC, making the two triangles similar, so we can use the ratio of equivalent sides to show that the length of XY does indeed give the value of tanq. In similar triangles, the sides grow in the same proportion, so we have the ratios XYAY = BCAC.
Substituting the values gives us tanq1 = sinqcosq, = which is the relationship that we know gives us the slope of the line. So we can measure the length XY to give the value for tanq. The diagram gives only an approximation, but it does explain why we call the third ratio the tangent.
The nice thing about approaching tangents in this way is that now you can look at what happens to the value of tanq as the rod moves around the circle - as it approaches the vertical (q = 90o), the value of tanq tends towards infinity, and at 90o the rod will lie parallel to the tangent on the y axis.
I have created an animated unit circle using PowerPoint so that you can see the changing values. There are interactive geometry packages that can be used for demonstration.
Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX