# 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(

*) cannot equal 2; and (5) be able to determine for what*

**x***cos(*

**x***) = -1 without using a GDC.*

**x**__4 questions__ - ‘accessible’ to ‘discriminating’

__4 questions__-

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

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

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

Test_SL_trig_functns_eqns_v1

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 **Worked solutions** for the Test_SL_trig_functns_eqns (above).

Test_HL_trig_functns_eqns_v1

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 **Worked solutions** for the Test_HL_trig_functns_eqns (above).