Solution Found!
Prove the sum of angle formula for the sine function by
Chapter 4, Problem 38E(choose chapter or problem)
PROBLEM 38E
Prove the sum of angle formula for the sine function by following these steps. Fix x.
(a) Let f(t):= sin (x + t). Show that f’’(t) + f(t) = 0, f(0) = sin x, and f ‘(0)= cos x.
(b) Use the auxiliary equation technique to solve the initial value problem y’’+ y = 0, y(0) = sin x, and y’(0)= cos x.
(c) By uniqueness, the solution in part (b) is the same as f (t) from part (a). Write this equality; this should be the standard sum of angle formula for sin (x + t).
Questions & Answers
QUESTION:
PROBLEM 38E
Prove the sum of angle formula for the sine function by following these steps. Fix x.
(a) Let f(t):= sin (x + t). Show that f’’(t) + f(t) = 0, f(0) = sin x, and f ‘(0)= cos x.
(b) Use the auxiliary equation technique to solve the initial value problem y’’+ y = 0, y(0) = sin x, and y’(0)= cos x.
(c) By uniqueness, the solution in part (b) is the same as f (t) from part (a). Write this equality; this should be the standard sum of angle formula for sin (x + t).
ANSWER:
SOLUTION
Step 1
In this problem, we are asked to prove the sum of angle formula for sine function.