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Find the measure, in degrees, of the smallest positive angle theta for which sin (3 theta) = cos (6 theta).

 May 23, 2022
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We can make use of the identity \(\cos \theta = \sin(90^\circ - \theta)\).

 

\(\sin 3\theta = \cos 6\theta\\ \sin 3\theta = \sin(90^\circ - 6\theta)\)

 

To get the smallest positive angle theta, we remove the sin from both sides (note that it does not always work, but if we assume that 0 degrees < 3 theta < 90 degrees and 0 degrees < 90 degrees - 6 theta < 90 degrees, then it works.)

 

\(3\theta = 90^\circ -6\theta\\ 9\theta = 90^\circ\\ \theta = \boxed{10^\circ}\)

 May 23, 2022

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