CPhill

Note....   sin (-θ)  =  -sin(θ)

And  tanθ =√15 / 15  =   1 / √15 

So

-sin (θ)  = -1/4  ⇒   sin (θ) = 1/4

And

tan (θ)  =  sin (θ)  / cos(θ)       rearrange as

cos (θ)  =  sin (θ)  / tan (θ)   =   (1/4)  / (1/ √15)   =  (1/4) * (√15 / 1)  = √15 / 4  ⇒  C

2 

We know the side adjacent to the angle  = 6

And we are looking for the hypotenuse = board length

The trig function that relates the adjacent and hypotenuse is the cosine

So

cos (55°)  =  6 / hyp      rearrange as

hyp   =   6 / cos (55°)  ≈ 10.5 ft

Note  .... cos (-θ) = cos(θ)     and  sin (-θ)  =  -sin(θ)

So....the next step should be

cos(θ) / - sin (θ)  =  -1 / tan (θ)   ⇒  "B"

1  Here, AB is the hypotenuse and BC is the opposite side

So....the trig function that relates  opp/hyp   =  cosine

So

cos (22°)  =  BC / 18     multiply both sides by 18

18 *cos (22°)  = BC ≈  16.7

P.S.  -  drawing a picture might help you with these, Cerenetie

We know the adjacent side to the angle  = 40

And we are looking for the opposite side to the angle = tree height

The trig function that relates the opposite / adjacent  =  tangent

So

tan ( 50°)  =  opp / 40        multiply both by 40

40 tan (50°)  =  opp   = tree height  ≈  47.7 ft

LOL!!!!

Don't worry about it, ANSP.....

We probably won't PERISH   ( at least for awhile, I hope  !!!   )

The distance from a point ( m, n)   to  line in the form  Ax + By + C  = 0  is given by :

l A(m) + B(n)  + C l

________________

√ [ A^2 + B^2]

So....  A = 2,   B = 1  and C = - 5

Thanks, tertre!!!!

That IS easier to use.....!!!

We have these points on the graph   (0,0)  and ( 3, - 2)

The slope  is   [ -2 - 0 ] / [ 3 - 0 ]   =   -2/3

So.....using the slope and (0,0).....we have the equation

y = (-2/3) ( x - 0) + 0

y = - (2/3)x        multiply through by 3

3y  = -2x      add 2x  to both sides

2x + 3y   = 0

5x + 8y = -9

In the form Ax  + By  = C

The slope is  -A / B

So....the slope of L1  is   -5/8

And L2  is a perpendicular line to L1.....so.....it has a negative reciprocal slope  =  8/5

So.....the equation of L2   is

y =  (8/5) x - 10) + 10

y = (8/5)x - 16 + 10

y = (8/5)x  - 6

So

m + b   =   (8/5)  - 6   =   (8/5) - 30/5    =    - 22/5