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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

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

oh? Interesting

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  !!!

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

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).

oh nice

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

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

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

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.

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

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

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.

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.

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.

(a - b) / (a + b)  = 3 /5       cross-multiply

3 (a + b)  = 5 (a - b)

3a + 3b =  5a  - 5b

3b + 5b  =  5a  - 3a

8b  =  2a

4b  =  a

a / b  =  4

sin A = 2/3  

So tan   A =   2 / [ 3^2 - 2^2 ] =  2 / sqrt 5

So 

tan A =   2/sqrt 5

BD /AD  =  2/sqrt 5

BD / 24 = 2/sqrt 5

BD =  48/sqrt  5  

sin C  =1/4

tan C  =    1 / [ 4^2 - 1^2 ]  =  1/ sqrt 15

So

tan C = 1/sqrt 15

BD / DC =  1/sqrt 15

(48 /sqrt 5)  / DC  =  1/sqrt 15

So

DC  =  (48/sqrt 5) / (1/sqrt 15)

DC =   48 (sqrt 15 / sqrt 5 )

DC  =  48sqrt 3

OK.....!!!!

We noted that the pattern  repeated every  120   digits

So

5000 mod 120   = 80

Looking  at what I posted before.....the 80th digit  in the pattern  =     9

So.....the 5000th digit  =   9  

Sum   = 

first term  [  1 - common ratio ^n ]   / [ 1  -common  ratio]

Where 

first term  1/2

common  ratio  =  1/3

n  = the  number of  summed terms

So

(1/2) [ 1 - (1/3)^6] / [ 1  -1/3 ]   =

(3/4) [ 1 - (1/3)^6]  =

182 / 243

Call the total distance =  D

The time for  the  first part of  the  trip  =  (2/5)D / 60   =   D / 150

The time  for the second part of the trip   =   (3/5)D / R        where R is the rate we are looking  for

And  total  distance   /   total time =   avg rate

So

D /  [  D/150  + (3/5)D / R ]  =    50

1 /  [  1/150  + 3/ (5R) ] =  50

1 / [  R/ (150R) + 90 / (150R) ]    = 50

150R /  [  90 +R ]   = 50

3R  / [ 90 + R ]=   1

3R =    90 + R

2R =  90

R =  45 mph

CPhill, that post doesn't help.  Can't you just answer the question?

b =  (8 - 2a) / 3

For  b to be an integer, a must be   1 , 4 , 7 , 10 .... 

2b + 30  =      2 ( b + 15)   =   2 [ (8 -2a) / 3  + 15 ]  =   2 [ 53 - 2a] / 3

Divisors  =    1 , 2 ,  6

Thank you so much!

See Alan's answer here :

https://web2.0calc.com/questions/please-help_63707