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Solution 4.4:6a

From Förberedande kurs i matematik 1

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If we move everything over to the left-hand side,

sinxcos3x−2sinx=0

we see that both terms have sinx as a common factor which we can take out,

sinx(cos3x−2)=0.

In this factorized version of the equation, we see the equation has a solution only when one of the factors sinx or cos3x−2 is zero. The factor sinx is zero for all values of x that are given by

x=n(n is an arbitrary integer)

(see exercise 3.5:2c). The other factor cos3x−2 can never be zero because the value of a cosine always lies between −1 and 1, which gives that the largest value of cos3x−2 is −1.

The solutions are therefore

x=n(n is an arbitrary integer).
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