CPhill

47 + 2* 360    =  767°

47  -   360    =  -313°

Second one

-135°   =  360 - 135   =  225°

So...we have   (3, 225°)

This one is easy.....it is a 45° angle in the 3rd Quadrant  

The sine and cosine are equal here....so the x,y coordinates will be the same

cos 225  =  x / 3

3cos 225  = x

3 ( -1 /√2)  = x

-3/√2  = x   = y

So

(x , y )   =  ( -3/√2 , -3/√2 )

First one

This is a little tricky.......r is always positive.....so....we need to add 180° to 120°

(-3, 120°)  =  ( 3, 300°)

So

cos 300 =  x / 3

3cos300 = x  

3 (1/2)  =  x  

1.5  = x

And

sin300 = y / 3

3sin300 = y

3 * -√3/2   = y  

-(3/2)√3   = y

-3√3 / 2   = y

So

(x , y )   =   ( 1.5, -3√3 / 2 )

2.Let $C$ be a point not on line $AE$ and $D$ a point on line $AE$ such that $CD \perp AE.$ Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE.$ If $AB = 4,$ $CD = 8,$ and $AE = 5,$ then what is the length of $CE?$

                    C

                           B

              4

                     8

  A                D                                   E

                       5

We have two triangles ...  ABE  and CED

Angle ABE = Angle CDE

Angle BEA = Angle CED

Therefore.....by AA congruency....

Triangle ABE  is similar to Triangle  CDE

So

AE / AB   =  CE / CD

5 / 4   =  CE / 8        cross-multiply

8 * 5 /  4   = CE

40 / 4   =  CE

10  = CE

\(BD = k \sqrt{2}\)

1. In triangle $ABC$, $AB = 3$, $BC = 4$, $AC = 5$, and $BD$ is the angle bisector. If $BD = k \sqrt{2}$, then find $k$.

A

3           5

B       4           C

The bisector will divide AC  into  a ratio  of 3 : 4   =  AD : CD

Therefore.....AD  =  (3/7)*5  =    15/7

The sine of BAC = 4/5

So  the cosine of BAC  =  √ [ 5^2 - 4^2 ] / 5  =   3/5

Using the Law of Cosines

BD^2   = AB^2 + AD^2  - 2 (AB * AD)cos BAC

BD^2 =  3^2  + ( 15/7)^2 - 2(3 * 15/7) (3/5)

BD^2  = 9 + 225/49 - 2(3 * 3 * 3) / 7

BD^2 =  9 +  225/49 - 54/ 7

BD^2  =   [ 441 + 225 - 378] / 49

BD^2  = 288 /49

BD = √288/ 7  =   (12/7) √2

So....   k  = (12/7)

sin (5x) / 2x    multiply top/bottom by 5

5sin (5x) / [ 5 * 2x]

(5/ 2)  sin (5x) / [ 5x]

As x  ⇒  0  the limit   of  sin (5x) / [5x}  = limit of  sin (0) / 0   =  1

So....5/2    is correct !!!!

tan x              sinx / cosx                      sin x * cos x            sin x

_____    =      ____________ =          ___________ =      _____

xsecx               x  [ 1 /  cos x]               x * cos x                   x

And the limit  of the last function as x ⇒ 0   =   1     !!!!

Correct on both counts !!!

(x^2-5x-24)/(x-8)  =

(x + 3) ( x - 8) / ( x - 8)  =

x + 3

This will be a line with a "hole" at x = 8

We can make this coninuous by a piecewise function

f(x)   =  (x^2-5x-24)/(x-8)    if x  not equal to 8

                  11                        if x   =  8

Q

P    6√2         R

Since angle Q  = angle R.......this is a 45 - 45 - 90 right triangle

Angle P is right

And the sides opposite the other two angles are equal....so... PR  = QP

But these are the leg lengths

And the area of a right triangle  =

[ Product of leg lengths ] / 2   =  

[ 6√  2  * 6√2 ] / 2  =

[ 36 * 2] / 2  =

36 units^2

Since 7  is greater than pi....the absolute value   of   pi - 7  =   7 - pi

So  we have

[ pi  - (7 - pi ) l   =

l  2pi - 7 l

And 2pi < 7   ...... so the absolute value =

7 - 2pi

The distance from the beach to the island  will form the altitude, A, of a triangle

Let x represent the distance  that the person with the 30 degree line of sight is from where the altitude intersects the base

So...the distance that the person with the 40 degree line of sight to this intersection point is 50 -x

So   we have this

tan 30  = A/ x       

Rearrange as  [ x  =  A / tan 30  ] ⇒

[ x =    A / (1/√3)  =  √3A ]

And we have that

tan 40 = A / (50 - x)       sub in for x  and we get that

tan 40  = A / ( 50 - √3A)

(50 - √3A) * tan 40  = A

50sin 40 - √3A tan40  = A

50tan 40   =   √3Atan 40 + A          factor out A on the right

50tan 40  =  A ( √3 tan 40 + 1)      divide both sides by  (sin 40 + 1)

A = 50tan 40 / [√3 tan 40 + 1 ] ≈   17.1 (m)

EDIT TO CORRECT AN ERROR!!!