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More trig problems

I still have some more trig problems left that I have trouble figuring out.

1.       Graph the function y =  .5tan(.5x-(pi/4)) for -pi ≤ x ≤ 4pi.

2.       Graph the functions y = 1/2x and y = sinx on the same            axis over the interval 0 ≤ x  ≤ 4pi.

3.       Find all positive values of x less than 360 degrees that           satisfy the equation x = arcsin(-(sqrt3)/2).

4.       Evaluate tan(cos^ -1(.8)) Assume the angle is acute.

5.       Evaluate sin( arctan((sqrt3)/3)).

6.       Evaluate sin^ -1(-(sqrt2)/2) in radians.

7.       State the range of y = tan^ -1x.

8.       Graph the function y = arccos x.

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Answer by: Ganesha

Best Answer
1)
y =0.5tan(0.5x  -  pi/4) ,  -pi <=  x  <=4pi
Putting t = 0.5x-pi/4, the trange becomes -pi/2-pi/4  <=  t  <= 2pi-pi/4   ==>
-3pi/4 <=  t  <=  -7Pi/4
I have done the work in Ms Exel . Please  try to plot as the values are given.

Sl =  a Pi/4 values  = b t =a*b 0.5tan(t) Remarks
-3 0.785398163 -2.356194489 0.500000001 The actual value of y is  0.5
-2 0.785398163 -1.570796326 -629012585.9  
-2.01 0.785398163 -1.578650308 63.66067471 Therefe is infinite discontinuity for t=-2
-1.99 0.785398163 -1.562942344 -63.66066182 Therefe is infinite discontinuity for t=-2
-1 0.785398163 -0.785398163 -0.5  
0 0.785398163 0 0  
1 0.785398163 0.785398163 0.5  
1.99 0.785398163 1.562942344 63.66066182 Therefe is infinite discontinuity for t=2
2.01 0.785398163 1.578650308 -63.66067471 Therefe is infinite discontinuity for t=2
3 0.785398163 2.356194489 -0.500000001  
4 0.785398163 3.141592652 -7.94897E-10 The actual value  of y is zero.
5 0.785398163 3.926990815 0.499999998 The actual value of y is 0.5
5.99 0.785398163 4.704534996 63.66064894 Therefe is infinite discontinuity for t=+6
6.01 0.785398163 4.72024296 -63.6606876 Therefe is infinite discontinuity for t=+6
7 0.785398163 5.497787141 -0.500000003  
         
         
There is small error of the orde 1.0*10^-1 in the value of  y as the value of pi is approximated  to 3.141592654


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Answer by: Ganesha

First Answer
4)
tan(cos inverse(.8)) =  sqrt(1-.82)/.8  =.6/.8 =0.75
5)
sin(arc tan(sqrt(3)/2)  = sin (arcsin (sqrt(3/((sqrt3)2+32) = sqrt(3/12) = 1/2
6)
sin inverse(-sqrt2)/2)  =  45 deg = P/4 rad
7)
- P/2 to +P/2

Sorry!,  I am sleepy. Let us see tomorrow for the rest of the problems.


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Answer by: Ganesha

3) x=arc(sin(-sqrt3)/2) ==>
sinx = -[  sqrt3]/2.
sin is negative in 3rd and 4th quadrants.
x = 240   or 300  degrees  .  Or,   x =4pi/3  or  5pi/3  radians

8)
Please plot values  x and   y= arccos x from below:

x acosx
-1 3.141592654
-0.9 2.690565842
-0.8 2.498091545
-0.7 2.346193823
-0.6 2.214297436
-0.5 2.094395102
-0.4 1.982313173
-0.3 1.875488981
-0.2 1.772154248
-0.1 1.670963748
0 1.570796327
0.1 1.470628906
0.2 1.369438406
0.3 1.266103673
0.4 1.159279481
0.5 1.047197551
0.6 0.927295218
0.7 0.79539883
0.8 0.643501109
0.9 0.451026812
1 0
0.9 0.451026812
0.8 0.643501109
0.7 0.79539883
0.6 0.927295218
0.5 1.047197551
0.4 1.159279481
0.3 1.266103673
0.2 1.369438406
0.1 1.470628906
0 1.570796327
-0.1 1.670963748
-0.2 1.772154248
-0.3 1.875488981
-0.4 1.982313173
-0.5 2.094395102
-0.6 2.214297436
-0.7 2.346193823
-0.8 2.498091545
-0.9 2.690565842
-1 3.141592654
x values are -1<=x<=1 
again 1 >=x>=-1. 


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Answer by: junie

idk this srry
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Answer by: Ganesha

2)
The soltion is in Ms Exel. The graph may be plotted by you as the required values are  provided. You can approximate the values to two decimal places for the purpose of plotting.

n x= n* (4pi/48) y   =  (1/2)*x  y   = sin(x)  
0 0 0 0  
1 0.261799388 0.130899694 0.258819045  
2 0.523598776 0.261799388 0.5  
3 0.785398164 0.392699082 0.707106781  
4 1.047197551 0.523598776 0.866025404  
5 1.308996939 0.65449847 0.965925826  
6 1.570796327 0.785398164 1  
7 1.832595715 0.916297857 0.965925826  
8 2.094395103 1.047197551 0.866025404  
9 2.356194491 1.178097245 0.707106781  
10 2.617993878 1.308996939 0.5  
11 2.879793266 1.439896633 0.258819045  
12 3.141592654 1.570796327 -4.10207E-10 zero
13 3.403392042 1.701696021 -0.258819046  
14 3.66519143 1.832595715 -0.5  
15 3.926990818 1.963495409 -0.707106782  
16 4.188790205 2.094395103 -0.866025404  
17 4.450589593 2.225294797 -0.965925826  
18 4.712388981 2.356194491 -1  
19 4.974188369 2.487094184 -0.965925826  
20 5.235987757 2.617993878 -0.866025403  
21 5.497787145 2.748893572 -0.707106781  
22 5.759586532 2.879793266 -0.499999999  
23 6.02138592 3.01069296 -0.258819044  
24 6.283185308 3.141592654 8.20414E-10 zero
25 6.544984696 3.272492348 0.258819046  
26 6.806784084 3.403392042 0.500000001  
27 7.068583472 3.534291736 0.707106782  
28 7.330382859 3.66519143 0.866025404  
29 7.592182247 3.796091124 0.965925827  
30 7.853981635 3.926990818 1  
31 8.115781023 4.057890511 0.965925826  
32 8.377580411 4.188790205 0.866025403  
33 8.639379799 4.319689899 0.70710678  
34 8.901179186 4.450589593 0.499999999  
35 9.162978574 4.581489287 0.258819044  
36 9.424777962 4.712388981 -1.23062E-09 zero
37 9.68657735 4.843288675 -0.258819046  
38 9.948376738 4.974188369 -0.500000001  
39 10.21017613 5.105088063 -0.707106782  
40 10.47197551 5.235987757 -0.866025404  
41 10.7337749 5.366887451 -0.965925827  
42 10.99557429 5.497787145 -1  
43 11.25737368 5.628686838 -0.965925826  
44 11.51917306 5.759586532 -0.866025403  
45 11.78097245 5.890486226 -0.70710678  
46 12.04277184 6.02138592 -0.499999999  
47 12.30457123 6.152285614 -0.258819044  
48 12.56637062 6.283185308 1.64083E-09 zero
         
The value at conlumn C and D are the required y values.  
The  4pi's are devide into 48 intervals , the pi value is   
taken3.141592654. Each interval is also equal to 15 degrees.

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