Question 1.

A hallway of width 6 feet meets a hallway of width 9 feet at right angles. Find the length of the longest pipe that can be carried level around this corner.
(Book's Hint: If "L" is the length of the pipe, show that L = 6cscO + 9csc((Pi/2) - O) Where O is the angle between the pipe and the wall of the narrower hallway)
Solution 1.



Question 2.

There is a single 21-foot vertical palm tree growing in the middle of a large flat desert island. On a certain day, the sun will rise at 6:00 *** and set at 6:00 PM. At noon on that day the sun will be directly overhead and the trunk of the palm tree will cast no shadow. A traveler lay down the night before this special day, and was awakened in the morning when the sun's rays reached his eyes, which were 13 feet due west from the tree. To the nearest minute at what time did he awake?

Solution 2.

Let A be the angle between sunray and palm tree.
At 6am when sun rise up from horizon, A=180 degree; at 12am when the sun is shining right above the tree, A= 0 degree.
When the sun travels from 6 *** to 12 ***(6 hrs), it traces the curve= quarter of the circle= 90 degree
so the rate of the sun tracing the curve = 90/6=15 degree/hr= .25degree/min
let h=21feet (height of the tree) r=13feet
so A=arctan(13/21) =32 degree
but the angle that the sun has traveled is 90-32=58 degree
T(time)= 58/.25=232 mins=3hrs and 52 mins
so at 9:52 *** this person was waken up by the sun.

Question 3.

Find out information about Trigonomic Identities.

Solution 3.

Trigonometric Identities:
cscA=1/sinA secA=1/cosA tanA=sinA/cosA cotA=cosA/sinA = 1/tanA sin[SUP]2[/SUP] A+cos[SUP]2[/SUP] A=1
1+tan[SUP]2[/SUP]A=sec[SUP]2[/SUP]A 1+cot[SUP]2[/SUP] A=csc[SUP]2[/SUP]A
sin(-A)=-sinA cos(-A) = cos A
tan (-A)= -tan A sin(pi/2 - A) = cos A
cos(pi/2- A)= sinA tan (pi/2-A)= cotA


Question 4.

Find the measure of the marked acute angle to the nearest degree.

Solution 4.

ArcTan(9/2) = 77.47 degree. or 77 degree as requested to the nearest degree.



Question 5.


Verify the identity tanx+coty/tanxcoty=tany+cotx

Solution 5.

This is true only when x+y=pi/2
tanx+ coty/(tanxcoty)=tanx+ 1/tanx=tanx+cotx
=tan(pi/2-y)+cotx
=tany+ cotx