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Find all acute angles x such that (sin2x)*(sin3x) = (cos2x)*(cos3x). If the sum of these angles can be written in the form aπ/b in lowest terms, what is a + b?

 

I am stuck on the fact that I can get one solution, which is x = 18 deg, or π/10 radians. In addition, x = -18 deg or -π/10 radians is also a viable solution, but when summed, the solutions cancel out and form literally zero. So, is the answer one or zero? Am I wrong completely? Is there another solution I'm missing? Are negative angles not to be conidered in this problem? I am very confused, if someone can help ASAP that would be great.

 May 5, 2022
 #1
avatar+3587 
0

Your solutions are correct, but you don't need to add them. The sum a+b is  neither 1 or 0, but it's 11: Your solution is π/10=1π/10, so a=1 and b=10.

 

Another solution would be π/2. With this solution the sum a+b would be 3.

 May 5, 2022
edited by Probolobo  May 5, 2022
 #2
avatar+29 
+1

I see but the question clearly states that "the sum of these angles can be written in the form...", so I feel like I do need to sum them up.

ineedhelponmath  May 5, 2022
 #3
avatar+3587 
+1

I can't read lol. "sum of these angles" - of course you need to sum the answers.

With the two answers π/10 and π/2, the sum is π/10+π/2=π/10+5π/10 = 6π/10 = 3π/5. So the sum a+b is 3+5=8.

Probolobo  May 5, 2022
 #4
avatar+29 
+1

Okay but what about the negative acute angles? Wouldn't that completely negate the sum? Or would negative angles just not come into play?

ineedhelponmath  May 5, 2022
 #5
avatar+29 
+3

Also thanks for being so responsive at such a late hour!

ineedhelponmath  May 5, 2022
 #6
avatar+3587 
+2

I think I got it now. Didn't notice the word "acute", probably because I usually don't do math in english.

An acute angle is an angle x with 0

 

There are still more solutions: all angles of the form \(\frac{(2n-1)\pi}{10}\) (with n a natural number). So with n=1, we get your solution π/10, and with n=2, we get 3π/10. n=3 gets us π/2, wich is no acute angle, so π/10 and 3π/10 are the only solution to your question. That let's us solve the rest: The sum is π/5, so a+b=6.

Probolobo  May 5, 2022
edited by Probolobo  May 5, 2022
edited by Probolobo  May 5, 2022
 #7
avatar+29 
+1

Awesome, this makes so much sense now. Thank you so much!

ineedhelponmath  May 5, 2022
 #8
avatar+29 
+1

Wait, how is it angle x with zero???

ineedhelponmath  May 5, 2022
 #9
avatar+3587 
+1

Somehow the forum wouldn't let me edit my answer: 

An acute angle is an angle x with \(0

Also, it's not late for me since I'm from Germany - it's actually 0800 :D

Probolobo  May 5, 2022
edited by Probolobo  May 5, 2022
 #10
avatar+29 
+1

oh nice

ineedhelponmath  May 5, 2022
 #11
avatar+3587 
+2

You got another pn, somehow I can't edit my answer to be complete. What I wanted to say is that an acute angle is between 0 and 90° (strictly, so 0° and 90° aren't acute).

Probolobo  May 5, 2022
 #12
avatar+29 
+2

that makes so much more sense now. Thanks for clarifying. Ty Probolobo!!

ineedhelponmath  May 5, 2022
 #17
avatar+3587 
+1

Of course π/10+3π/10 =2π/5, not π/5, sorry! So a+b=2+5=7, as CPhill said. 

Probolobo  May 5, 2022
 #18
avatar+29 
+1

Yep got it. Thanks everyone!

ineedhelponmath  May 5, 2022
 #13
avatar+122385 
+2

sin 2x sin 3x  = cos 2x cos 3x

 

This might be bettter solved graphically...

 

https://www.desmos.com/calculator/qkom6uy1lg

 

Notice that the graph (of the solution points) are completely symmetric around  the y axis

 

So....the sum of  the angles   will  be 0

 

In other words, if    (x , y)  is on the graph  then so is (-x , y)....the sum of the x values "cancel" to 0  !!!

 

 

cool cool cool

 May 5, 2022
edited by CPhill  May 5, 2022
 #14
avatar+29 
+1

oh? Interesting

ineedhelponmath  May 5, 2022
 #15
avatar+29 
+1

But then what does that make a + b equal to? Would it just be one then?

ineedhelponmath  May 5, 2022
 #16
avatar+122385 
+2

I misread the problem ....it asks for  the acute angles 

 

By the graph  the acute angles are

 

pi/10 , 3pi/10 

 

The sum of these  =   4pi/10  =   2pi/ 5

 

a + b = 7

 

https://www.desmos.com/calculator/sgeoiv21y3

 

cool cool cool

CPhill  May 5, 2022
edited by CPhill  May 5, 2022
edited by CPhill  May 5, 2022

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