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**Unformatted text preview: **− 2 y = cos(3x) and y = 2 − cos(x) 5. While we could approach cos(3x) = cos(5x) in the same manner as we did the previous two
problems, we choose instead to showcase the utility of the Sum to Product Identities. From
cos(3x) = cos(5x), we get cos(5x) − cos(3x) = 0, and it is the presence of 0 on the right
hand side that indicates a switch to a product would be a good move.6 Using Theorem 10.21,
we have that cos(5x) − cos(3x) = −2 sin 5x+3x sin 5x−3x = −2 sin(4x) sin(x). Hence,
2
2
the equation cos(5x) = cos(3x) is equivalent to −2 sin(4x) sin(x) = 0. From this, we get
sin(4x) = 0 or sin(x) = 0. Solving sin(4x) = 0 gives x = π k for integers k , and the solution
4
to sin(x) = 0 is x = πk for integers k . The second set of solutions is contained in the ﬁrst set
of solutions,7 so our ﬁnal solution to cos(5x) = cos(3x) is x = π k for integers k . There are
4
π
π
π
π
eight of these answers which lie in [0, 2π ): x = 0, π , π , 34 , π , 54 , 32 and 74 . Our plot of the
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graphs...

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