MATH Week 10. Ferenc Balogh Winter. Concordia University

Transcription

1 MATH 20 - Week 0 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson All figures and videos are made using MAPLE and ImageMagick-convert.

2 Overview Trigonometric Identities - Section 7. Trigonometric Identity vs Trigonometric Equation The Starting Point: Fundamental Trigonometric Identities Simplifying Trigonometric Expressions Proving Trigonometric Identities

3 Trigonometric Identity A trigonometric identity is an equation of trigonometric expressions in θ valid for all values of θ. For example sin 2 θ + cos 2 θ is valid for all θ. Trigonometric Equation A trigonometric equation is an equation of trigonometric expressions in θ valid for some particular values of θ. For example sin θ + cos θ is valid for θ 0 but it is invalid for θ π 4.

4 Fundamental Trigonometric Identities Reciprocal Identities csc x sin x tan x sin x Pythagorean Identities sec x cot x sin x cot x tan x sin 2 x + cos 2 x tan 2 x + sec 2 x + cot 2 x csc 2 x Even-Odd Identities sin( x) sin x cos( x) tan( x) tan x

5 Cofunction Identities ( π ) ( π ) ( π ) sin 2 x tan 2 x cot x sec 2 x csc x ( π ) ( π ) ( π ) cos 2 x sin x cot 2 x tan x csc 2 x sec x Although these are considered to be fundamental identities, it is more convenient to prove them later on.

6 Example. Simplify the trigonometric expression sin t + cos t cot t. Solution. sin t + cos t cot t sin t + cos t cos t sin t sin2 t + cos 2 t sin t sin t csc t.

7 Example. Simplify the trigonometric expression Solution. sin x + sin x +. sin x + sin x + ( + ) + sin2 x sin x( + ) + cos2 x + sin 2 x sin x( + ) + sin x( + ) sin x csc x.

8 How to prove a trigonometric identity? We transform one side of the equation into the other side by using a sequence of steps. Hints Start with one side (take the more complicated one). Use the fundamental identities and perform algebraic manipulations. In case of an emergency, write all expressions in terms of sines and cosines only. PRACTICE!!!

9 Example. Prove the identity sec x sec x rewriting it in terms of sines and cosines. Solution. The LHS is LHS sin 2 x. sec x sec x cos 2 x cos2 x cos 2 x sin 2 x RHS.

10 Example. Verify the identity sec x + tan x + sec x tan x 2 sec x. Solution. LHS sec x + tan x + sec x tan x sec x tan x + sec x + tan x (sec x + tan x)(sec x tan x) 2 sec x sec 2 x tan 2 x 2 sec x 2 sec x RHS.

11 Example. Verify the identity sin 4 t cos 4 t sin 2 t cos 2 t. Solution. LHS sin 4 t cos 4 t sin 4 t (cos 2 t) 2 sin 4 t ( sin 2 t) 2 sin 4 t ( 2 sin 2 t + sin 4 t) sin 4 t + 2 sin 2 t sin 4 t + 2 sin 2 t (sin 2 t + cos 2 t) + 2 sin 2 t sin 2 t cos 2 t + 2 sin 2 t sin 2 t cos 2 t RHS.

12 Example. Verify the identity Solution. + tan 2 u tan 2 u cos 2 u sin 2 u. LHS + tan2 u tan 2 u + ( sin u cos u ( sin u ) 2 ) 2 cos u cos 2 u+sin 2 u cos 2 u cos 2 u sin 2 u cos 2 u cos2 u + sin 2 u cos 2 u cos2 u + sin 2 u cos 2 u cos 2 u sin 2 u cos 2 u sin 2 u cos 2 u sin 2 u RHS.

13 Example. Verify the identity Solution. tan v tan v + tan v tan v LHS tan v tan v + cos v cos v + sin 2 v cos v + cos v cos v sin2 v cos v cos v + cos v sin 2 v + cos v + cos v

14 Solution (cont). RHS tan v tan v cos v cos v cos v cos v cos v cos v cos v cos v cos v cos v + cos v + cos v cos 2 v ( + cos v) sin 2 v ( + cos v) cos v sin 2 v + cos v.