# Trigonometry

#### Syllabus Content - SL & HL

Although trigonometry historically developed from the use of triangles in applications such as navigation, surveying, map-making and architecture, it is important for students to be aware that there are essentially two different approaches to trigonometry (and trigonometric functions). One is a more empirical-based approach where trigonometry is mostly about functions of angles (navigation, surveying, etc), whereas another approach is viewing trigonometry as concerning functions of real numbers. Students will have (at least should have) been introduced to trigonometry before appearing in a Math SL or Math HL class (see trigonometry section in Prior Learning Topics) - and almost certainly their prior study of trigonometry will have been from a 'functions of angles' approach. I suggest that the teaching of trigonometry in the Math SL and HL courses should be split into two different teaching units - one that considers trig functions as functions of real numbers and the second unit with trig functions interpreted as functions of angles. Although students are more familiar (and comfortable) with trigonometry being applied to triangles and other geometric shapes, I believe that it is best to teach the trig functions of real numbers unit first. One reason for this is simply that it is an approach that is less familiar to students.

#### Teaching Approaches

The teaching of the Circular Functions & Trigonometry Topic (or just simply 'Trigonometry') can be approached from two different perspectives - trigonometric functions considered as (1) functions of angles, or (2) functions of real numbers. Of course, an angle - whether given in degree measure or radian measure - is certainly a real number so trigonometric functions are always functions of real numbers. However, when trigonometric functions are applied in a geometric context (e.g. finding the missing side or angle of a triangle) the domain of the relevant trig function will be a real number that is the measure of an angle. Whereas, there are many applications of trig functions (periodic phenomena such as wave motion, oscillating pendulums, natural life cylces, etc) where the function domain has no reference to angles or triangles.

Thus, the Trigonometry Topic can be divided into two sections. The division between the two sections is not entirely clear and distinct - and there is some overlap. However, I have found that it helps students have a more organized approach to this area of the syllabus. I feel it also helps to teach and present the material in a more logical manner. The approach that considers trig functions in a geometric context is what I refer to as triangle trigonometry, and the approach where trig functions are more primarily considered in a non-geometric context I describe as trigonometric functions & equations.

Here are links to three different kinds of Trigonometry Unit Tests that are located elsewhere on this site. The first unit test covers triangle trigonometry and is suitable for both SL and HL. The 2nd and 3rd unit tests cover trigonometric functions and equations - one for SL and the other for HL. The 4th and 5th unit tests cover all of trigonometry and are most suitable for HL.  The 5th is a bit more challenging than the 4th test.
1.   Test on triangle trigonometry - SL & HL  (with solution key)
2.   Test on trigonometric functions & equations - SL  (with solution key)
3.   Test on trigonometric functions & equations - HL  (with solution key)
4.   Test1 on trigonometry (both triangle trig & trig functions) - HL  (with solution key)
5.   Test2 on trigonometry (both triangle trig & trig functions) - HL  (with solution key)

The triangle trigonometry section of the Topic includes syllabus items such as solution of triangles, area of a triangle, the cosine rule, the sine rule and - to some extent - circle geometry (arcs, sectors & segments). The trigonometric functions & equations section starts with arc lengths and radian measure (and some circle geometry) and then focuses on defintions of the trig functions in terms of the unit circle, graphs of trig functions, trig identities, solving trig equations and inverse trig functions (HL only). Since many students are quite familiar with using trig functions to solve for sides and angle of triangles including use of the cosine and sine rules, I think it it best to teach the trig functions & equations section first. This gives students a solid theoretical understanding of trig functions and their non-geometic applications with which students are usually less familiar. It also provides a nice opportunity to teach students the nature of circular functions (or 'wrapping' functions) by means of the unit circle. By starting the Trigonometry Topic with a more functions-oriented approach this also connects nicely with the Functions & Equations Topic which will most likely have been taught just before teaching the Trigonometry Topic. Important features of functions and equations - domain, range, composite functions, inverse functions, transformations of graphs, solving equations, related calculator skills, etc - that were treated in Topic 2 will again be very important in this syllabus Topic.

## Selected Pages

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### sine cosine tracer13 May 2018

Geogebra applet using coordinates of points on the unit circle to trace out one period of sine curve or cosine curve.
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