- Er 1 In This Question You Will Be Using The Following Trigonometric Identities Cos A Sin A 1 1 Cos A 3 Cos 1 (43.56 KiB) Viewed 30 times
ER 1. In this question, you will be using the following trigonometric identities: cos a + sin’a = 1 (1) cos(a + 3) = cos a cos B - sin a sin B (2) sin(a +B) = sin a cos 3 + cos a sin B (3) where a, ß ER. You do not need to prove these identities. You may also use without proof the fact that the set cos sina is exactly the set of unit vectors in R2. Now for any real number a, define cosa-sina RA sina cosa (a) Prove that for all a, B ER, R, R3 = Ra+B (b) Using part (a), or otherwise, prove that Ra is invertible and that R' = R-a, for all a € R. (c) Prove that for all a € R and all x, y € R?, (R,x). (R.y) = xy (d) Suppose A is a 2 x 2 matrix such that for all x, y € R?, (Ax). (Ay) = xy Must it be true that A = Ra, for some a € R? Either prove this, or give a counterexample (including justification). -