About cookies

The NCETM site uses cookies. Read more about our privacy policy

Please agree to accept our cookies. If you continue to use the site, we'll assume you're happy to accept them.

 

Personal Learning Login






Sign Up | Forgotten password?
 
Register with the NCETM

Secondary Magazine - Issue 114: Sixth Sense


Created on 04 September 2014 by ncetm_administrator
Updated on 17 September 2014 by ncetm_administrator

 

Secondary Magazine Issue 114'SIX' by Richard Gray (adapted), some rights reserved
 

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 right-angled 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.

Unit circle on trigonometric chart

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.

unit circle on trigonometric chart

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

\small 1 < \frac{\Theta }{sin \Theta } < \frac{1}{cos \Theta }

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 \small \sqrt{a^{2}+b^{2}}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

 
 

 

 
 
 
 
Download the magazine as a PDF
 
Secondary Magazine Archive
 
Magazine Feed - keep informed of forthcoming issues
 
Departmental Workshops - Structured professional development activities
 
Explore the Secondary Forum
 
Contact us - share your ideas and comments 
 

 


Comment on this item  
 
Add to your NCETM favourites
Remove from your NCETM favourites
Add a note on this item
Recommend to a friend
Comment on this item
Send to printer
Request a reminder of this item
Cancel a reminder of this item

Comments

 


17 September 2014 08:35
We've added PDF versions of the diagrams - click the image to view. Thanks for the feedback!
16 September 2014 19:31
Unfortunately, I can't really see the diagrams, they are small and illegible when zoomed in on.
By christinapt
         Alert us about this comment  
Only registered users may comment. Log in to comment