97. The Ultimate Sinusoidal Equation It is an interesting fact that any sinusoid can be written in the form
y=asin[b(x−H)]+k
where both a and b are positive numbers.
(a) Explain why you can assume b is positive. [Hint: Replace b by −B and simplify.]
(b) Use one of the horizontal translation identities to prove that the equation
y=acos[b(x−h)]+k
has the same graph as
y=asin[b(x−H)]+k
for a correctly chosen value of H. Explain how to choose H.
(c) Give a unit circle argument for the identity sin(θ+π)=−sinθ. Support your unit circle argument graphically.
(d) Use the identity from (c) to prove that
y=−asin[b(x−h)]+k,a>0
has the same graph as
y=asin[b(x−H)]+k,a>0
for a correctly chosen value of H. Explain how to choose H.
(e) Combine your results from (a)-(d) to prove that any sinusoid can be represented by the equation
y=asin[b(x−H)]+k
where a and b are both positive.