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.

ineedhelponmath May 5, 2022

#1**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.~~

Probolobo May 5, 2022

#2**+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**+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**+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

#6**+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

#11**+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**+2 **

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

ineedhelponmath
May 5, 2022

#13**+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 !!!

CPhill May 5, 2022

#15**+1 **

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

ineedhelponmath
May 5, 2022