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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,

 Dec 31, 2020
 #1
avatar+94 
+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. :))))))

 

=^ ._.^=

 Dec 31, 2020
 #2
avatar+114171 
+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°

 

cool cool cool

 Dec 31, 2020
edited by CPhill  Dec 31, 2020
 #3
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+1

Thanks guys, I'm so grateful you 2 are both on the forum.

 Dec 31, 2020

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