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,

cos(x-90)=sin(x)

cos(7*theta)=sin(3*theta),

therefore

x-90=7*theta

x=3*theta

then x=-22.5, but that is negative and the answer must be positive,

Guest Dec 31, 2020

#1**+1 **

Hello Guest!

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. :))))))

=^ ._.^=

catmg Dec 31, 2020

#2**+1 **

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°

So

5x = 45°

x = 9°

CPhill Dec 31, 2020