Suppose that 𝑎 is a fixed number, and consider the function 𝑔(𝑥)=sin(𝑥−𝑎)+cos(𝑥+𝑎)

Question

Suppose that 𝑎 is a fixed number, and consider the function  𝑔(𝑥)=sin(𝑥−𝑎)+cos(𝑥+𝑎) 

Answer the following questions. Show and explain your work.

(a) If 𝑎≠𝜋/4 what is the period of the function 𝑔(𝑥)? Does it depend on 𝑎?

(b) Graph the function 𝑔(𝑥) when 𝑎=𝜋/4 What do you see? Provide an explanation to your observation (the explanation can be either geometric or algebraic).

(c) Show, using trigonometric identities, that for any value of 𝑎, the graph of 𝑔(𝑥) is simply a vertical scaling of the curve 𝑦=sin𝑥+cos𝑥, possibly with a reflection about the x-axis.

Answer 1

Correction: Critical values in quadrants 1 and 3, not 1 and 2.

Answer 2

Pauline,

I suggest to start with c) Use the identities

sin(x-a)=sin(x)cos(a)-cos(x)sin(a) (2)

cos(x+a)=cos(x)cos(a)-sin(x)sin(a) (3) so g(x) becomes

𝑔(𝑥)=sin(𝑥−𝑎)+cos(𝑥+𝑎) =[sin(x)+cos(x)]*[cos(a)-sin(a)](3)

which is the scaling of the [sin(x)+cos(x)] function by a positive factor [when cos(a))>sin(a)] and negative in the opposite case

From (3) it becomes evident that for a=pi/4 g(x)=0 for any x

That'd answer the b) part

Finally, from 3) it follows that the period of g(x) is 2pi