Trig functions, equations & identities

Teacher notes

The trigonometry syllabus content can essentially be split into two parts (with some overlap) of triangle trigonometry and trigonometric functions, equations & identities. Triangle trigonometry involves functions of angles whereas the other approach to trigonometry (trig functions, equations & identities) deals with trigonometric functions defined in terms of a real number that is the length of an arc along the unit circle where we are mostly interested in: graphs of trigonometric functions, solving trigonometric equations, inverse trigonometric functions and applying trigonometric identiies. Students need to use and understand radian measure, and to be aware whether or not the domain of a trig function is in degrees or radians.

A key aspect of solving trigonometric equations is whether or not an exact solution is required. It is very important that students get sufficient practice in solving a range of trig equations where any solutions need to be expressed exactly - with an emphasis on equations that require the solution(s) to be expressed in radian measure.

Although a range of solution strategies should be discussed and practiced, the most important technique with which students never seem to get enough practice is substituting an appropriate trigonometric identity in order to facilitate the solution of a trig equation. For example, a good starter exercise in this regard is the following question:


Although this is not a difficult equation to solve, students do need to come up with a reasonable strategy before they set out their work - and then make suitable adjustments while carrying out their work. They need to: (1) recognize that a substitution is required since there are two different trig functions in the original equation; (2) choose an appropriate substitution; (3) recognize that the equation is quadratic in terms of cos(x) and that it can be solved by factorising; (4) be aware that cos(x) cannot equal 2; and (5) be able to determine for what x cos(x) = -1 without using a GDC.

4 questions - ‘accessible’ to ‘discriminating’

download: 4_Qs_trig_functns_eqns_identities_1_with_answers_v2  

accessible SL question

moderate SL / accessible HL question

discriminating SL / moderate HL question

discriminating HL question

Answers

 ♦ teaching materials

AA_SL_3.8_trig_eqns1_v1  laugh
Set of 10 trigonometric equations to be solved. No GDC for Qs 1-7, GDC allowed for Qs 8-10. Worked solutions for Qs 1-4, 8, 9; answers (and hints) for Qs 5-7, 10 are attached.

Quiz_HL_trig_functns_eqns_v1 laugh
HL quiz on trigonometric functions, equations & identities. 7 questions - 4 with no GDC, and 3 with GDC allowed. Worked solutions available below.

Quiz_HL_trig_functns_eqns_v1_SOL_KEY laugh
Worked solutions for the HL quiz (above) on trigonometric functons, equations & identities.

Test_SL_trig_functns_eqns_v1 laugh
SL test on trigonometric functions & equations with 9 questions: 1-6 No GDC; 7-9 GDC allowed. Worked solutions available below.

Test_SL_trig_functns_eqns_v1_SOL_KEY  laugh
Worked solutions for the Test_SL_trig_functns_eqns (above).

Test_HL_trig_functns_eqns_v1 laugh
HL test on trigonometric functions & equations with 11 questions: 1-7 No GDC; 8-11 GDC allowed. Worked solutions available below.

Test_HL_trig_functns_eqns_v1_SOL_KEY  laugh
Worked solutions for the Test_HL_trig_functns_eqns (above).

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