2 103 and most recently 106 we solved some basic

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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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