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1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 (1) %3D cos(a + B) = cos a cos 3 – sin a sin 3 sin(a + B) (2) sin a cos 3 + cos a sin 3 (3) %3D 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 :a € R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a - sin a Ra sin a cos a (a) Prove that for all a, ß E R, R,R3 = Ra+8 (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 E R and all x, y e R², (R.x) · (Ray) = x · y

1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 (1) %3D cos(a + B) = cos a cos 3 – sin a sin 3 sin(a + B) (2) sin a cos 3 + cos a sin 3 (3) %3D 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 :a € R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a - sin a Ra sin a cos a (a) Prove that for all a, ß E R, R,R3 = Ra+8 (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 E R and all x, y e R², (R.x) · (Ray) = x · y

Trigonometry (MindTap Course List)
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
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Can you please help me so work out this question? I found this on Textbook but there is only answer, so I can not clearly understand.

1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 cos(@ + B) sin(a + B) (1) (2) (3) cos a cos B – sin a sin 3 sin a cos 3 + cos a sin 3 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 R?. Now for any real number a, define cos a - sin a cos a Ra sin a (a) Prove that for all a, 3 e R, R R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R' = R-a, for all a E R. (c) Prove that for all a E R and all x, y € R?, (Rax) · (Ray) = x • y (d) Suppose A is a 2 × 2 matrix such that for all x, y € R², (Ах) (Ау) —D х у Must it be true that A = Ro, for some a e R? Either prove this, or give a counterexample (including justification).
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 cos(@ + B) sin(a + B) (1) (2) (3) cos a cos B – sin a sin 3 sin a cos 3 + cos a sin 3 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 R?. Now for any real number a, define cos a - sin a cos a Ra sin a (a) Prove that for all a, 3 e R, R R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R' = R-a, for all a E R. (c) Prove that for all a E R and all x, y € R?, (Rax) · (Ray) = x • y (d) Suppose A is a 2 × 2 matrix such that for all x, y € R², (Ах) (Ау) —D х у Must it be true that A = Ro, for some a e R? Either prove this, or give a counterexample (including justification).
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