CPhill

x^2  +  y^2  =  4      (1)

(x - 5)^2  + (y - 5)^2 = 40   ⇒   x^2 + y^2  - 10x - 10y + 25 + 25 =  40  ⇒

4 - 10x - 10y  + 50  =  40 ⇒   -10x -10y =  -14  ⇒    5x + 5y  = 7  ⇒   5y = 7 -5x ⇒  y = (7 -5x) / 5    (2)

Sub (2)  into (1) for y

x^2 +  [ (7- 5x)/5)^2  = 4

x^2  +  [ 49 - 70x + 25x^2 ] / 25   =  4          multiply through by 25

25x^2  + 25x^2 - 70x  + 49   = 100

50x^2 - 70x - 51  =   0

The approx solutions to  this  are    x ≈ -.528  and  x ≈ 1.92

And  when   x ≈ -.528   y ≈1.92

And when x  ≈ 1.92   y ≈ -.528

The slope  is   [   1.92  - - .528 ] / [ -.528 - 1.92]   =  [1.92 + .528 ] /  - [ 1.92 + .528 ]  =   -1

We can expect   five  composite rolls for  every prime  roll....so....

5 * 185 =  925   composites

Angle  B  in the smaller( isosceles) triangle =   (180 -70) / 2  =  55°

So we can find the  base of the larger triangle as

base  /sin70  = 27/sin55

base =  27*sin 70 / sin 55  ≈  31

So

x =  31/sin27  ≈  68.3°

a)   the tangent is negative in this quadrant     (true)

b)   the sine is positive in this quadrant  so the cosecant   is also positive  (false)

I'm assuming that the height of th balcony is  4m

So....the distance between them  is  given by

4 /tan (52°)   +  4/tan 47°)   ≈   6.86 ft

A  =   (5/18)*180  =   50°

B =    (6/18) * 180 = 60°

C  =   (7/18)* 180 =  70°

By the Law of Sines

a /sin 50  = b / sin 60                 c / sin 70  = b / sin 60

a / b =  sin 50 /sqrt(3)/2             c/b = sin 70 / sqrt (3)/2 

a =  b  [ 2sin 50  / sqrt 3]            c =   b [ 2sin 70 / sqrt 3 ]

a  +  b +  c  =   180

b* 2sin 50 /sqrt 3    + b  +  b [ 2sin 70 /sqrt 3 ]   =  18

b  [  2sin 50 /sqrt 3  + 1  + 2sin70/sqrt 3 ] =  18

b =  18  /  [ 2sin50/sqrt 3  + 1 + 2sin 70 / sqrt 3 ]  ≈  6.06

a = 6.06 [ 2 sin 50 / sqrt 3]  ≈ 5.36

c =  6.06 [2 sin 70 /sqrt 3 ]  ≈ 6.58

Area =  sqrt  [ 9  (9 - 6.06) (9 - 5.36) (9 - 6.58)  ]  ≈   15.267  units

1/3 =  10/30  =  

11/12 =   27.5 / 30 

So

[ 27  -11 ] + 1  =    17 integer values

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

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  !!!

(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