Question

The function f is defined as
f(x) = -2sinx ; -π≤x≤-π/2
= asinx + b ; -π/2<x<π/2
= cosx ; π/2≤x≤π
If f(x) is continuous in the interval -π≤x≤π fined the value of ‘a’ and ‘b’.

Answer 1

Since the f(x) is to be continuous, then end point of one piece must be the starting point of the second piece, etc.
So the link values are -π/2 and π/2
from y = -sinx, we get (-π/2, 2)
and from y = cosx, we get (π/2, 0)
so the middle function y = asinx + b must contain the points (-π/2,2) and (π/2,0)
2 = asin(-π/2) + b
2 = -a + b , #1

0 = asin(π/2) + b
0 = a + b , #2

add #1 and #2
2 = 2b
b = 1

back into #2
0 = a + 1
a = -1

so the second leg of the function is
y = -sinx + 1