
= Let cos(6) sin(0) B - sin() cos() and 0 << 27 (i) Calculate the eigenvalues of B. Hence prove that the modulus of the eigenvalues is equal to one. (ii) Calculate the eigenvectors of B.
tan 0 24) Simplify: seco A)sin 0 B)cos e C)csc 0 D)sin 0 – csc O E) csc 0 – sin 0
3 Given sin osesan and sin B -7 37 25 <B< 27. Find cos(0 + B).
Establish the identity 1 - cos 0 sin 0 + sin 0 1 - cos 0 = 2 csc 0. Which of the following shows the key steps in establishing the identity? 1 - cos e sin 0 ОА. + sin e 1 cos e 1 - cos e B + sin e 1 - cos 0 sin e (1 - cos 0)2 + sine 2 = 2 csc 6 sin 0(1 - cos ) cOS (1 - cos 02...
3. If T2 = r3 cos(0) sin(d) and v2 = sin(0) cos(O)f + r sin(0)θ + r2 sin(d)φ compute the following (a) ▽T, (b) ▽.
establish the identity
Establish the identity. cos 0 sin = sin 0 - cos 0 - 1- tan 0 - 1- coto Write the left side in terms of sine and cosine. cos 0 sin o -1- Write each term from the previous step as one fraction. cos?o sin 0 - cos 0 (List the terms in the same order as they appear in the original list.) Add the fractions from the previous step. (Do not simplify.) cos 0 -...
[6] sin 2B given sec B - 3 cos 2B and & sin >0. In what quadrant does 2B terminate? 7 5 [7] Verify the identity: 2 csc A sin A 1 + cos A + 1 + cos A sin A
NOTE: Very useful trigonometric identities are these: sin(A B)-sin A cos B sin B cosA, cos(A +B)-COSA cos B-sin A sin B 32. (Bonus problem) A periodic function g(x)is defined on one period like this: g(x).0' on 1<x<0, and it equals x on 0<<1 (a) Give a labeled sketch of the graph of g(x), let's say from-1.5 to 3.5 (b) Give labeled sketches of, the graphs of g (x) and g(x) (i.e, the even and odd parts ofg).
that 20 Suppose 0 <0 < and sin 0 = }. Evaluate: (a) csce (b) sece (c) cote
The spin operator along an arbitrary direction = (sin 0 cos 0) is defined as sin 0 sin o cos (1.18) h where S 2 _ (a) Find the eigenstates of the above spin operator and show that their eigenvalues are + 2 h h would be measured to have S (b) Find the probability that the state S =+ 2 where n' 2 would be measured to have S 2 (c) Find the probability that the state S =...