Mathematics Grade 12 (4-1) Trigonometric Identities . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation ML3.PA4 Grade 12 ()Learning Outcome : : :: Assessment Criteria : . . . ML3.PA4.1 Identify and use basic trig identities to find trig values

ML3.PA4.2 Use basic trig identities to simplify and rewrite trig expressions . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 Trigonometric identities Earlier in the course you met the following trigonometric identities: sin tan (cos 0) 1 cos

sin2 cos2 1 2 We can write these identities in terms of sec , cosec and cot . Using 1 So 1 = 1 cos cot = = sin tan sin cos cos cot sin (sin 0) . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Trigonometric identities 2 PA4.1 sin2 cos2 1 Dividing 2 through by cos2 gives sin2 cos2 1 + cos2 cos2 cos2 tan2 +1 sec 2 Dividing 2 through by sin2 gives sin2 cos2 1 + 2 2 2 sin sin sin 1+ cot 2 cosec 2 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA4.1 Trigonometric Identities Quotient Identities sin tan cos cos cot sin Reciprocal Identities 1 1 cos sin sec csc 1 tan cot

Pythagorean Identities sin2+ cos2 = 1 tan2+ 1 = sec2 cot2+ 1 = csc2 sin2= 1 - cos2 tan2= sec2- 1cot2= csc2- 1 cos2 = 1 - sin2 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 Trigonometric Identities EVEN-ODD IDENTITIES sin sin cos cos cosec cosec sec sec tan tan cot cot Cofunction Identities sin cos

2 cos sin 2 tan cot 2 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation Using the identities you now know, find the trig value. 2.) If cos = 3/5, find csc. PA4.1 1.) If cos = 3/4, find sec

sin 2 cos 2 1 2 3 sin 2 1 5 25 9 sin 2 25 25 16 sin 2 25 4 sin 5 1 1 5 sin

4 4 5 1 1 4 sec cos 3 3 4 csc . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 Using the identities you now know, find the trig value. 3.) sin = -1/3, find tan tan 2 1 sec 2

tan 2 1 ( 3) 2 tan 2 2 tan 2 8 tan 2 8 4.) sec = -7/5, find sin . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 : : 189 : : : 190 : : 26 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA4.1 Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. a) cos sin tan sin cos sin cos sin2 cos cos cos 2 sin2 cos 1 cos sec

cot 2 2 1 sin b) cos 2 sin 2 2 cos cos 2 1 2 2 sin cos 1 sin2 csc 2 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simplifying Trigonometric Expressions PA4.1 Identities can be used to simplify trigonometric expressions. Simplify. (1 + tan x)2 - 2 sin x sec x )c csc x tan x cot x )d . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation Simplifying Trigonometric Expressions PA4.1 Simplify each expression.

1 sin cos sin 1 sin sin cos 1 sec cos 1 sin x cos x sin x cos x cos x cos x

sin x sin x 1 cos2 x sin 2 x sin x sin x cos2 x sin 2 x sin x 1 csc x sin x . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation Simplifying Trigonometric Expressions PA4.1 Simplifying trig Identity cosx

Example1: simplify tanx sin tanxx cosx cos x tanxcosx = sin x . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation Simplifying Trigonometric Expressions PA4.1 Simplifying trig Identity 1 sinx x cos x 1 = sin x

cos x = sec x csc x 1 cos sec x csc 1x sin x Example2: simplify = tan x . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation Simplifying Trigonometric Expressions PA4.1 Simplify:

= cot x (csc2 x - 1) Factor out cot x = cot x (cot2 x) Use pythagorean identity Simplify = cot3 x . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 Simplifying Trigonometric Expressions Simplify: = sin x (sin x) + cos xUse quotient identity cos x cos x 2 = sin x + (cos cos x) x Simplify fraction with LCD cos x = sin2 x + cos2x

Simplify numerator cos x = 1 Use pythagorean identity cos x = sec x Use reciprocal identity . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation Simplifying Trigonometric Expressions !Your Turn PA4.1 Combine fraction Simplify the numerator Use pythagorean

identity Use Reciprocal Identity . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation PA4.1 : : 194 : : : 195 : : 27 . Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation