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Pre-Calc Exam Notes 130

Pre-Calc Exam Notes 130 - 130 Chapter 6 Additional Topics...

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130 Chapter 6 Additional Topics §6.1 Example6.2 Solve the equation 2cos 2 θ 1 = 0. Solution: Isolating cos 2 θ gives us cos 2 θ = 1 2 cos θ = ± 1 radicallow 2 θ = π 4 , 3 π 4 , 5 π 4 , 7 π 4 , and since the period of cosine is 2 π , we would add 2 π k to each of those angles to get the general solution. But notice that the above angles differ by multiples of π 2 . So since every multiple of 2 π is also a multiple of π 2 , we can combine those four separate answers into one: θ = π 4 + π 2 k for k = 0, ± 1, ± 2, ... Example6.3 Solve the equation 2 sec θ = 1. Solution: Isolating sec θ gives us sec θ = 1 2 cos θ = 1 sec θ = 2 , which is impossible. Thus, there is no solution . Example6.4 Solve the equation cos θ = tan θ . Solution: The idea here is to use identities to put everything in terms of a single trigonometric function: cos θ = tan θ cos θ = sin θ cos θ cos 2 θ = sin θ 1 sin 2 θ = sin θ 0 = sin 2 θ + sin θ 1 The last equation looks more complicated than the original equation, but notice that it is actually a
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