Question

Let x and y be arbitrary angles.
a. Use the result of Exercise 1a (see answer for 1a below) to prove that sin x cos x >= (-1/2)
b. Use the facts that sin (-y) = -sin y and cos (-y) = cos y to prove that sin x cos x <= 1/2.
c. Prove that sin x + cos x <= sqrt (2)


1a) (sin x+ cos x)^2= (sin x)^2 + (cos x)^2 + 2 sin x cos x
since (sin x)^2 + (cos x)^2 =1
so the answer is 1 + 2 sin x cos x

Answer 1

aliter to 1a:
sin2x >= -1 ( bcos sin(A) is in the range [-1,1] )
2*sinx*cosx >= -1
sinx cosx >= -0.5

Answer 2

1c.
sin(x+45)<=1
expanding:
(sinx+cosx)/(sqrt2) <= 1
sinx+cosx<=sqrt2

Answer 3

what do u mean "aliter to 1a"?

Answer 4

are you there?