CPhill

I think this is supposed to be

2sin^2 x  = sin x

2*sin^2 x - sin x  =  0              factor out sin x

sin x ( 2sin x - 1)  =  0

sin x  = 0       and this occurs when x = 0   and x = pi

And

2sin x - 1  = 0

2sin x  = 1

sin x  =1/2    and this occurs when x = pi/6  and x  = 5pi/6

We have that

arc length  = r * theta.....so

(2pi).9  = 8 * theta       divide  both sides by 8

(2pi) / ( 9 * 8)  = theta

(2/72)pi = theta

pi/36  =  theta  (in rads)

I get a slightly different result

The number of digits written down =

9  + 2(99 - 10 + 1)  + 3(999 - 100 + 1)  =

9 + 2 ( 90)  + 3(900)  =

2889

The number of  7's  that are written down  =

10*10  + 10(10)  + 100  =   300

So.....the probability  of  selecting a "7"  =   300 / 2889  = 100/963

LOL!!!!....I'm still wondering myself    !!!!

x^2 + ( y + 5)^2  = 25

x = r cos θ

y = r sin θ

So we have

(r cos θ)^2  +  (r sin θ + 5)^2  =  25

r^2 cos^2 θ  +  r^2 sin^2 θ  + 10r sin θ + 25  = 25

r^2  ( cos ^2 θ  + sin^2 θ ) + 10rsin θ =  0

r^2  ( 1)  +  10r sin θ  =  0      

r^2  + 10r sin θ  =  0           divide out r

r +10 sin θ =  0

r = -10 sin θ

3 hrs 45 min  =  3 + 3/4  hrs  =  15/4  hrs

Distance =  rate * time

80 km/ hr  * (15/4) hr   = (80*15)/4 km  = (20 * 15) km =  300 km

Here's the graph : 

https://www.desmos.com/calculator/bwhu2zaivl

This is  the regular cosine graph with an amplitude of 3,  horizontally compressed by a factor of 2  and shifted down one unit

x = 3√2 cos (-pi/6)  = 3√2 * √3/2  =  (3/2)√6

y = 3√2  sin (-pi/6)  = 3√2 * (-1/2)  = -3√2 / 2 = -(3/2)√2

So

(x , y)   =     ( (3/2)√6 ,  -(3/2)√2  )

r + r cosθ = 6

r = √[x^2 + y^2]

cos θ =  x/r 

So we have that

√[x^2 + y^2]  +  r (x/r)  =  6

√[x^2 + y^2] + x  = 6

√[x^2 + y^2]  =  6 - x        square both sides

x^2  + y^2  = x^2 - 12x + 36

y^2  = -12x + 36

12x = -y^2 + 36

x = -y^2/12 + 3