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**Unformatted text preview: **lled the phase of the sinusoid. Since our interest
φ
in this book is primarily with graphing sinusoids, we focus our attention on the horizontal shift − ω induced by φ. The proof of Theorem 10.23 is a direct application of Theorem 1.7 in Section 1.8 and is left to the
reader. The parameter ω , which is stipulated to be positive, is called the (angular) frequency of
the sinusoid and is the number of cycles the sinusoid completes over a 2π interval. We can always
ensure ω > 0 using the Even/Odd Identities.7 We now test out Theorem 10.23 using the functions
f and g featured in Example 10.5.1. First, we write f (x) in the form prescribed in Theorem 10.23,
πx − π
π
π
+ 1 = 3 cos
x+ −
+ 1,
2
2
2
so that A = 3, ω = π , φ = − π and B = 1. According to Theorem 10.23, the period of f is
2
2
φ
2π
2π
= π/2 = 4, the amplitude is |A| = |3| = 3, the phase shift is − ω = − −π/2 = 1 (indicating
ω
π/2
f (x) = 3 cos 7 Try using the formulas in Theorem 10.23 applied to C (x) = cos(−x + π ) to see why we need ω &...

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