Find the measure, in degrees, of the smallest positive angle $\theta$ for which $\sin 3 \theta = \cos 7 \theta.$
My go at it:
Using the theorem for sin and cosine relationship,
then x=-22.5, but that is negative and the answer must be positive,
Formulas are quite tricky to remember and this happens to me all the time.
cos(90° – x) = sin x and cos(x-90) !=sin(x)
You're 100% on the right track, but used the wrong formula.
Following what you did, \(sin(3\theta) = cos(90-3\theta) = cos(7\theta).\)
\(cos(90-3\theta) = cos(7\theta) \)
For this statement to be true, the difference between the angles inside need to be a multiple of 360.
For simplicity, I'm just going to set them as the same because it doesn't matter in this case.
\(90-3\theta = 7\theta \)
\(90 = 10\theta\)
\(\theta = 9 \)
One thing to note is that angle x = angle x + 360.
Thus, getting an answer of -22.5 is perfectly reasonable, you'd just have to do -22.5 + 360.
I hope this helped. :))))))
sin (3x ) = sin ( 5x - 2x)
cos (7x) = cos ( 5x + 2x)
sin (3x) = cos (7x)
sin (5x - 2x) = cos (5x + 2x)
sin 5x cos2x - sin 2xcos 5x = cos5xcos2x - sin5xsin2x rearrange as
sin 5x ( cos 2x + sin2x) = cos 5x( sin 2x + cos 2x)
sin 5x = cos 5x
Note that sin (theta) = cos (theta) at 45°
5x = 45°
x = 9°