371 problems 8, 24, 64, 92


Page 371 Problem 8

ArcTan( - Sqrt[3]/3)   = -Pi/6

                        |   3
              ----------+-------------
                        |\ 30   |
                        | \     |
                        |  \    | -Sqrt[3]
                        |   \   |
                   2 Sqrt[3] \  |
                        |     \ |
                        |      \|
                        |


Page 371 Problem 24 


Express theta in terms of x using the inverse trigonometric

functions ArcSin, ArcCos, ArcTan, ArcSec




                               /|
                              / |
                             /  |
                            /   | 3
                         x /    |
                          /     |
                         /t   . |
                        / . '  _| 2
                       /___s__|_|
                            
                      Sqrt[x^2-25]

            

              t = ArcTan[5/Sqrt[x^2-25]]-ArcTan[2/Sqrt[x^2-25]]; 

 

Page 371 Problem 64 


Evaluate the integral

                  t=Pi/2    Sin[t]
                   INT  ------------- dt
                   t=0         2 
                          1+Cos [t]

                             |t=Pi/2
             -ArcTan[Cos[t]] |
                             |t=0


             -ArcTan[0] + ArcTan[1]

                        = Pi/4.

Page 371 Problem 92 


A man on a dock is pulling in a rope attached to

a rowboat at a rate of 5 feet per second.  If the man's hands

are 8 feet higher than the point where the rope is attached

to the boat, how fast is the angle of depression of the rope

changing when there are still 17 feet of rope out?

        __________________
       |\ a
       | \
       |  \
       |   \
       |    \
     8 |     \
       |      \
       |       \
       |        \
       |         \
       |        a \
       |___________\
            x


        t = ArcCtn[x/8]

                     1/8 dx/dt
        da/dt = - ------------ 
                     1+(x/8)^2

               |           1/8 5
         da/dt |      - --------------  =   - 40/353 radians/sec
               |x=17      1+(17/8)^2