1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 cos(a + B) sin(a + B) = cos a cos B – sin a sin B = sin a cos B + cos a sin B (1) (2) (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 sin a is exactly the set of unit vectors in R?. Now for any real number a, define cos a – sin cos a sin a Ra (a) Prove that for all a, B ER, RaR8 = Ra+B (b) Using part (a), or otherwise, prove that R. is invertible and that R 1 = R, for all a E R. %3D (c) Prove that for all a E R and all x, y E R², (Rax)· (Ray) = x y (d) Suppose A is a 2 × 2 matrix such that for all x, y E R?, (Ax) · (Ay) = x y Must it be true that A = Ra, for some a E R? Either prove this, or give a counterexample (including justification). (e) Let B be any 2 × 2 matrix. %3D
1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 cos(a + B) sin(a + B) = cos a cos B – sin a sin B = sin a cos B + cos a sin B (1) (2) (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 sin a is exactly the set of unit vectors in R?. Now for any real number a, define cos a – sin cos a sin a Ra (a) Prove that for all a, B ER, RaR8 = Ra+B (b) Using part (a), or otherwise, prove that R. is invertible and that R 1 = R, for all a E R. %3D (c) Prove that for all a E R and all x, y E R², (Rax)· (Ray) = x y (d) Suppose A is a 2 × 2 matrix such that for all x, y E R?, (Ax) · (Ay) = x y Must it be true that A = Ra, for some a E R? Either prove this, or give a counterexample (including justification). (e) Let B be any 2 × 2 matrix. %3D
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 65E
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage