Sixth Sense
Key representations in Level 3 Mathematics – The Unit Circle
The development and redefinition of trigonometric functions as functions of real numbers from the definitions of trigonometric ratios as scale factors between the side lengths of rightangled triangles is one of the ‘big ideas’ in Level 3 Mathematics.
The unit circle is a key visual representation in understanding this conceptual development. It allows the extension of the definition of all six trigonometric functions firstly to any angle in degrees, then to any angle in radians and then as functions of any real number, as well as providing a clear visual representation of all six trigonometric functions as displacements, and triangles which help establish the Pythagorean identities, some key trigonometric inequalities and the small angle approximations prior to differentiation of the trigonometric functions.
It is powerful when used in conjunction with trigonometric graphs (another key visual representation in trigonometry), and indeed, can be used dynamically to generate the trigonometric graphs themselves.
click image to download a larger version of the diagram (PDF)
In the diagram above, if angle SOP = θ (measured anticlockwise from Ox, at the moment either in degrees or radians) then:
 P is the point with coordinates (cos θ, sin θ) i.e. the displacement (i.e. the directed or signed distance) OQ represents cos θ and the displacement QP represents sin θ.
 R is the point with coordinates (1, tan θ) i.e. SR represents tan θ as a displacement along the tangent line to the unit circle at (1, 0).
 T is the point with coordinates (cot θ, 1) i.e. VT represents cot θ as a displacement along the tangent line to the unit circle at (0, 1).
 Additionally, OR represents sec θ as a displacement along the extended diameter OP and OT represents cosec θ as a displacement along the same extended diameter OP. A line that cuts a circle at two points is called a secant line; hence sec θ and cosec θ.
Note also that the triangles ΔOQP, ΔOSR and ΔOUT are similar, where it can be seen that
 sec θ/(1)= 1/cos θ, cosec θ/(1) = 1/sin θ and sin θ/cos θ = tan θ/(1) = 1/cot θ.
Using Pythagoras’ Theorem in each of these same triangles gives
 (cosθ)²+ (sinθ)²= 1, 1 + (tanθ)² = (secθ)², and (cotθ)²+ 1 = (cosecθ)².
It is also worth noting that if angle SOP = θ (now measured in radians) and arc length SP = s, then θ = s and P is both (cos θ, sin θ) and (cos s, sin s) i.e. we can regard both θ and s as real numbers and we have defined the trigonometric (or circular) functions of any real number (as both θ and s can take any real value), rather than being limited to functions of angles.
click image to download a larger version of the diagram (PDF)
The unit circle representation works perfectly for angles of any size and can be used to verify the results and statements above for angles in any of the four quadrants. It’s well worth your students trying this for themselves for an angle in the fourth quadrant as shown in the diagram above. Even better, they can develop the representation in Geogebra (or Autograph) to revisit and use dynamically in class this term.
Finally, QP < arc length SP < SR so sin θ < θ < tan θ. Equivalently, dividing through by sin θ (which is positive for acute θ), we derive
This implies that, for small θ, sin θ is approximately equal to θ, because the centre fraction in the inequality is “sandwiched” between two expressions that tend to 1 as θ tends to 0. This also implies that tan θ is also approximately equal to θ (since tan θ = sin θ ÷ cos θ). These arguments can be made rigorous (and will be in higher study), but they are intuitively plausible and are accessible with the aid of the unit circle diagram. A similar argument starts with the observation that the area of the triangle OPQ is less than the area of the sector OPS, and both are less than the area of the triangle ORS: asking your students to develop this argument will be instructive and interesting for them.
The unit circle representation is a recurring representation throughout Level 3 Mathematics and can be easily scaled to any circle of radius r. Examples of its use can also be found in topics in Further Pure Mathematics such as polar coordinates and complex numbers, and in circular motion in Mechanics. Wherever it is a natural representation, trigonometric functions are likely to be helpful in formalising the mathematical description of the ideas or being explored. For example, considering the complex numbers a + bj and b + aj as points on a circle of radius helps students see the geometrical interpretation of the effect of multiplying of complex number by j: an interesting result, and also one that contributes towards the motivation for the introduction of j into the number system.
Further reading
Key Ideas in Teaching Mathematics: Research Based Guidance for Ages 9 – 19 (Anne Watson, Keith Jones and Dave Pratt) OUP 2013 Chapter 9: Moving To Mathematics Beyond Age 16
Image credit
Page header by Richard Gray (adapted), some rights reserved
