Unformatted text preview: = 3 cos(x) − 2
4. cos(3x) = 2 − cos(x) x
2 =1 Solution.
1. We resist the temptation to divide both sides of 3 sin3 (x) = sin2 (x) by sin2 (x) and instead
gather all of the terms to one side of the equation and factor.
3 sin3 (x) = sin2 (x)
3 sin3 (x) − sin2 (x) = 0
sin2 (x)(3 sin(x) − 1) = 0
Factor out sin2 (x) from both terms.
We get sin2 (x) = 0 or 3 sin(x) − 1 = 0. Solving for sin(x), we ﬁnd sin(x) = 0 or sin(x) = 3 .
The solution to the ﬁrst equation is x = πk , with x = 0 and x = π being the two solutions
which lie in [0, 2π ). To solve sin(x) = 1 , we use the arcsine function to get x = arcsin 1 + 2πk
or x = π − arcsin 1 + 2πk for integers k . We ﬁnd the two solutions here which lie in [0, 2π )
to be x = arcsin 3 and x = π − arcsin 3 . To check graphically, we plot y = 3(sin(x))3 and
2 and ﬁnd the x-coordinates of the intersection points of these two curves. Some
y = (sin(x))
extra zooming is required near x = 0 and x = π to verify that these two curves do in fact
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