Unformatted text preview: Graphs of the Trigonometric Functions 677 amplitude baseline period
The phase shift of the sinusoid is the horizontal shift experienced by the fundamental cycle. We
have seen that a phase (horizontal) shift of π to the right takes f (x) = cos(x) to g (x) = sin(x) since
cos x − π = sin(x). As the reader can verify, a phase shift of π to the left takes g (x) = sin(x) to
f (x) = cos(x). The vertical shift of a sinusoid is exactly the same as the vertical shifts in Section
1.8. In most contexts, the vertical shift of a sinusoid is assumed to be 0, but we state the more
general case below. The following theorem, which is reminiscent of Theorem 1.7 in Section 1.8,
shows how to ﬁnd these four fundamental quantities from the formula of the given sinusoid.
Theorem 10.23. For ω > 0, the functions
C (x) = A cos(ωx + φ) + B
• have period 2π
ω and S (x) = A sin(ωx + φ) + B
• have phase shift − • have amplitude |A| φa
ω • have vertical shift B a
In some scientiﬁc and engineering circles, the quantity φ is ca...
View Full Document