CPhill

period  = pi

y intercept  = 0

Plot points   ( -pi/4, -1) , (0,0) and ( pi/4  ,1)

Simplify as

x^2 + (4m - 2)x + (m + 6)  =  0

We have one root when  the discriminant  =  0

(4m - 2)^2 - 4 (1) ( m + 6)  = 0

16m^2 - 16m + 4 - 4m  - 24   = 0

16m^2 -20m - 20  =  0 

4m^2 - 5m - 5  = 0 

m =   [ 5 + sqrt [ 25 + 80 ] ] / 8  =    [ 5 + sqrt 105 ] / 8

Simplify as

x^2 + 12x + y^2 + 2y  =  10        complete the square on x, y

x^2 + 12x + 36  + y^2 + 2y + 1 =  10 + 36 + 1

(x + 6)^2  + (y + 1)^2  =  47

47 = r^2

Area of this  circle =   pi r^2  =  pi * 47  =   47 pi

Period  = the distance between  successive "peaks"  on the curve

x value at first peak  =  1

x value at second peak  = 5

Period =  5   -1   =   4

a)  3(8/10)  + 8(2/10)  =  [ 24 + 16 ] / 10  =   40 / 10   = 4

b) 3(8/11) + 8 (3/11)  =  [ 24 + 24 ]  / 11 =   48/11

c) 3(8/12) + 8(4/12)  = [ 24 + 32 ]  / 12 = 58/12 =  29/6 

d)  3 ( 8 / (10+n))  + 8 ( 2 + n) / (10 + n)  = 6

24 / (10+n)  + [ 16 + 8n ] / (10+n)  = 6

24 + 16 + 8n   =  6(10+ n)

40 + 8n  = 60 + 6n

2n  = 20

n = 10

e)   3 (8  / (10 + n)) + 8 ( 2 + n)  /(10+n)  =  7

24 + 16 + 8n  =  7(10 + n)

40 + 8n  = 70 + 7n

n  =  30

x^2 -10x + y^2 + 10y  =  -c

Complete the square on x,y

x^2 -10x + 25  + y^2 + 10y + 25 =  -c + 25 + 25

(x + 5)^2  + ( y -5)^2   =  50  - c

We want

50 - c   =  1^2

50 - c  =  1

c  = 50 - 1 =   49

First one

Altitude YW =  sqrt [ XZ^2  - (YZ /2)^2 ]  = sqrt  [12^2  - (6sqrt 3)^2  ] = sqrt [ 144  - 108] = sqrt [ 36] = 6

sin (YZW  )  =  YW / YZ  = 6 / 12   = 1/2

Draw  WN  where N is the tangent point of  the semi-circle to YZ

radius of  semi-circle / sin (YZW) = WZ / sin (WNZ)

r  / (1/2)  = (6sqrt 3) / sin 90°

r = (1/2) (6sqrt 3)  =  3sqrt 3  = sqrt (27)

Area of semi-circle   = (1/2)pi r^2   = (1/2)pi (27)  =  13.5 pi

a^2 - 9   = (a + 3) (a - 3)

3a^2 - 3a  =  3a ( a  - 1)

GCF  =  1

sin (arcos (-sqrt (3) / 2)) =  sin (5/6 pi)  =  1/2

arctan ( sin (-pi/2))  =  arctan (-1)  = 7pi/4  =  -pi/4

1/2 - pi/4  =

[ 2  - pi ] /  4

Second one

Law of Cosines

AC^2  = AB^2 + BC^2  - 2(AB * BC) cos ABC

30^2  = 25^2 + 25^2  - 2(25 * 25) cos ABC

[ 30^2 - 25^2 -25^2 ] /  (-2 * 25 * 25)   = cos ABC

cos ABC =  7/25

sin ABC   = sqrt [ 25^2  - 7^2 ] / 25 =  sqrt [ 576 ] / 25  =   24 / 25