Hi can i please get worked solutions to these questions? 1. In this question, you will be using the following trigonometric identities:cos? a + sin? a(1)(2)(3)1cos(a + B)sin(a + B)cos a cos B – sin a sin Bsin a cos B + cos a sin 3where a, B E R. You do not need to prove these identities. You may also use withoutproof the fact that the setcos a: αERsin ais exactly the set of unit vectors in R?.Now for any real number , define[cos a - sinaRa =sin aCOS a(a) Prove that for all a, B E R,RaR3 = Ra+B(b) Using part (a), or otherwise, prove that Ra is invertible and that R1all a E R.= R_a, for(c) Prove that for all a E R and all x, y E R²,(Rax) · (Ray) =x · y(d) Suppose A is a 2 x 2 matrix such that for all x, y e R?,(Ах) (Ау) — х уMust it be true that A = Ra, for some a e R? Either prove this, or give acounterexample (including justification).(e) Let B =be any 2 x 2 matrix.(i) Show that there are real numbers u11 and a such that[cos a= U11sin aHint: expressas a scalar multiple of a unit vector, and hence find anexpression for u11 in terms of a and c.(ii) Let a e R. Use the invertibility of R. to prove that there are uniqueU12, U22 E R such that[cos a= U12sin asin a+ U22Cos a(iii) Use parts (i) and (ii) to show that B can be expressed in the formB = RUfor some a E R and some upper-triangular matrix U.(iv) Suppose that B = RaU = RgV, where , B E R and U and V are upper-triangular. Prove that if B is invertible, then U = ±V.2

As per our guidelines we are supposed to solve only first three subparts of the problem. For the…

1. In this question, you will be using the following trigonometric identities: cos? cos(a + B) sin(a + B) a + sin? a = 1 (1) (2) cos a cos B – sin a sin ß sin a cos B + cos a sin B where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set cos a : α R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a – sin a sin a R. = COS Q (a) Prove that for all a, ß E R, RaR3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. = R_a, for (c) Prove that for all a E R and all x, y e R², (Rax) · (Ray) =x • y

1. In this question, you will be using the following trigonometric identities: cos? cos(a + B) sin(a + B) a + sin? a = 1 (1) (2) cos a cos B – sin a sin ß sin a cos B + cos a sin B where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set cos a : α R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a – sin a sin a R. = COS Q (a) Prove that for all a, ß E R, RaR3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. = R_a, for (c) Prove that for all a E R and all x, y e R², (Rax) · (Ray) =x • y

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter8: Complex Numbers And Polarcoordinates

Section8.4: Roots Of A Complex Number

Problem 42PS

See similar textbooks