Math C30
Trigonometric Identities
Instructional Notes

E.1

Note: While trigonometric identities are included in this course, it is not intended that they be taught in the traditional manner. Rather, it is recommended that the time line of two weeks be adhered to and that time be spent on applications of these identities to angle calculations in addition to verification. As well, verification exercises may be done as proofs, to reinforce the learning objectives of Concept A.

The reciprocal identities are those whose product is 1.

E.g.:	cos sec = 1	or	cos = 1/sec
	sin csc =1	or	sin = 1/csc
	tan cot = 1	or	tan = 1/cot

These identities can be shown to be true by the students, using a unit circle (x,y,r) or by employing the right triangle approach.

The students may be asked to use these identities to verify a statement, or to determine a specified value.

E.2

The quotient identities are usually thought of as those identities where two or more functions are divided to obtain another function. For our purposes, these are;

tan =	sin	and	cot =	cos
	cos			sin

Students can be asked to develop these quotient identities from the unit circle (x,y,r) or from right triangle trigonometry. They might work in small discussion groups to develop these. Once these are developed, the students can be given a set of exercises in which they are asked to verify statements or to determine specific values.

E.3

The Pythagorean identities can be developed from either the right triangle or the unit circle. The students could be given an opportunity to develop these on their own, or in small groups.

These identities are:

sin² + cos² = 1

1 + tan² = sec²

1 + cot² = csc²

Students should be asked to find alternative forms of each; such as,
sin² = 1 - cos² , in order to become familiar with some of the common variations of these identities.

Once students have developed these identities, a set of exercises on verification and determining specific values can be assigned.

E.4

The development of the addition/subtraction identities can be quite complex. It is suggested that the teacher model the development of the identity cos (ß - ) = cos ß cos + sin ß sin . This development can be found in a number of resource texts. It usually employs the distance formula.

When this identity has been developed, the others can be derived using this as a starting point. The other identities to be used are

cos (ß + ) = cos ß cos - sin ß sin

sin (ß + ) = sin ß cos + cos ß sin

sin (ß - ) = sin ß cos - cos ß sin

tan (ß + ) =	tan ß + tan
	1 - tan ß tan

tan (ß - ) =	tan ß - tan
	1 + tan ß tan

As an addition, the teacher may wish to use these formulas to show cos (-ß) = cos ß, tan (-ß) = - tan ß, and the like.

E.5

The teacher can have the students derive the double-angle identities by substituting into the addition identities and simplifying.

The double-angle identities to be done are
cos 2 = cos² - sin² or
= 1- 2 sin² , or
= 2 cos² - 1

sin 2 = 2 sin cos

tan 2 = (2 tan v)/(1- tan² )

The students can begin with

sin ( + ß) = sin cos ß + cos sin ß and substitute for ß which becomes

sin ( + ) = sin cos + cos sin
sin (2) = 2 sin cos .

The other double-angle identities can be shown in a like manner.

Once the identities have been proved, a set of exercises on applications of these identities may be assigned.

E.6

Students can be instructed to develop expressions for sin 3, 4, 5, etc., based upon their knowledge of the identity for sin 2 They can do the same for cos n and tan n or simply conjecture what might happen. They should test their conjecture.

The above will probably not have much meaning in itself, except for some intrinsic interest. Therefore, students should also attempt to graph sin , sin2, sin 3, sin 4, etc., for values of from 0^o to 360^o and compare the graphs.

The students may also explore the relationships of sin nf by preparing charts (tables of values) for f from 0^o to 360^o and observing the results.

The teacher may assign various groups a different method of exploring sin nf and have the groups compare their results. The presence of more than one depiction should help students understand the effect of a multiplier on . These can also be done for cos n, tan n, time permitting, or as a project outside class time.

E.7

In this section, students can be given the half-angle identities, and asked to use them in applications. These half-angle identities are: