Unformatted text preview: ave chosen to start at 2 , 2 taking
A = −2. In this case, the phase shift is 2 so φ = − 6 for an answer of S (x) = −2 sin 3 x − π + 1 .
Alternatively, we could have extended the graph of y = f (x) to the left and considered a sine
function starting at − 2 , 1 , and so on. Each of these formulas determine the same sinusoid curve
and their formulas are all equivalent using identities. Speaking of identities, if we use the sum
identity for cosine, we can expand the formula to yield
C (x) = A cos(ωx + φ) + B = A cos(ωx) cos(φ) − A sin(ωx) sin(φ) + B. 680 Foundations of Trigonometry Similarly, using the sum identity for sine, we get
S (x) = A sin(ωx + φ) + B = A sin(ωx) cos(φ) + A cos(ωx) sin(φ) + B.
Making these observations allows us to recognize (and graph) functions as sinusoids which, at ﬁrst
glance, don’t appear to ﬁt the forms of either C (x) or S (x).
Example 10.5.3. Consider the function f (x) = cos(2x) − 3 sin(2x). Rewrite the formu...
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