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**Unformatted text preview: **la for f (x):
1. in the form C (x) = A cos(ωx + φ) + B for ω > 0
2. in the form S (x) = A sin(ωx + φ) + B for ω > 0
Check your answers analytically using identities and graphically using a calculator.
Solution.
1. The key to this problem is to use the expanded forms of the sinusoid formulas and match up
√
corresponding coeﬃcients. Equating f (x) = cos(2x) − 3 sin(2x) with the expanded form of
C (x) = A cos(ωx + φ) + B , we get
cos(2x) − √ 3 sin(2x) = A cos(ωx) cos(φ) − A sin(ωx) sin(φ) + B It should be clear we can take ω = 2 and B = 0 to get
cos(2x) − √ 3 sin(2x) = A cos(2x) cos(φ) − A sin(2x) sin(φ) To determine A and φ, a bit more work is involved. We get started by equating the coeﬃcients
of the trigonometric functions on either side of the equation. On the left hand side, the
coeﬃcient of cos(2x) is 1, while on the right hand side, it is A cos(φ). Since this equation
is to hold for all real numbers, we must have8 that √ cos(φ) = 1. Similarly, we ﬁnd by
A
equating the coeﬃcients of sin(2x) that A sin(φ) = 3...

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