130Chapter 6•Additional Topics§6.1Example6.2Solve the equation 2cos2θ−1=0.Solution:Isolating cos2θgives uscos2θ=12⇒cosθ= ±1radicallow2⇒θ=π4,3π4,5π4,7π4,and since the period of cosine is 2π, we would add 2πkto each of those angles to get the generalsolution. But notice that the above angles differ by multiples ofπ2. So since every multiple of 2πisalso a multiple ofπ2, we can combine those four separate answers into one:θ=π4+π2kfork=0,±1,±2,...Example6.3Solve the equation 2 secθ=1.Solution:Isolating secθgives ussecθ=12⇒cosθ=1secθ=2,which is impossible. Thus, there is no solution.Example6.4Solve the equation cosθ=tanθ.Solution:The idea here is to use identities to put everything in terms of a single trigonometricfunction:cosθ=tanθcosθ=sinθcosθcos2θ=sinθ1−sin2θ=sinθ0=sin2θ+sinθ−1The last equation looks more complicated than the original equation, but notice that it is actually a
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