1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) cos a cos B – sin a sin B sin(a + B) 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 : αΕR sin a is exactly the set of unit vectors in R2. Now for any real number a, define cos a - sina Ra = sin a COs a (a) Prove that for all a, B E R, R R3 = Ra+8 (b) Using part (a), or otherwise, prove Ra is invertible and that R. = R-a, for all a E R. (c) Prove that for all a ER and all x, y e R?, (Rax) · (Ray) =x•y (d) Suppose A is a 2 x 2 matrix such that for all x, y € 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 x 2 matrix. cos a = u11 sin a (i) Show that there are real numbers u11 and a such that Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a e R. Use the invertibility of R. to prove that there are unique U12, U22 E R such that [cos a = u12 sin a sin a + U22 COs a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a e R and some upper-triangular matrix U. (iv) Suppose that B = RaU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) cos a cos B – sin a sin B sin(a + B) 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 : αΕR sin a is exactly the set of unit vectors in R2. Now for any real number a, define cos a - sina Ra = sin a COs a (a) Prove that for all a, B E R, R R3 = Ra+8 (b) Using part (a), or otherwise, prove Ra is invertible and that R. = R-a, for all a E R. (c) Prove that for all a ER and all x, y e R?, (Rax) · (Ray) =x•y (d) Suppose A is a 2 x 2 matrix such that for all x, y € 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 x 2 matrix. cos a = u11 sin a (i) Show that there are real numbers u11 and a such that Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a e R. Use the invertibility of R. to prove that there are unique U12, U22 E R such that [cos a = u12 sin a sin a + U22 COs a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a e R and some upper-triangular matrix U. (iv) Suppose that B = RaU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter5: Identities And Formulas
Section: Chapter Questions
Problem 2GP
Related questions
Question
100%
Part (d) and (e) are the most important, please solve the whole question if possible ?
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 8 steps with 8 images
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning