The function f(x)= sin x + cos x can be written in the form f(x)= A sin (Bx + C). Find A, B, and C.

Question

A table might help

Answer 1

Use the identity

 

sin(a + b) = sin(a) cos(b) + cos(a) sin(b)

 

Multiply and divide f(x) by the square root of the sum of the squares of the coefficients, which is √(1² + 1²) = √2

 

f(x) = sin(x) + cos(x) = √2 [sin(x)(1/√2) + cos(x)(1/√2)] 

 

1/√2 = cos(π/4) = sin(π/4) ⇒

 

f(x) = √2 [sin(x) cos(π/4) + cos(x) sin(π/4)]

 

f(x) = √2 sin(x + π/4) = Asin(bx + C)

 

A = √2

B = 1

C = π/4