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Check my answers please! This is the second part of my final and I just want to make sure I did everything correctly. 

1. Graph the function f(x) = -sec(x/4)+3 on the interval [-2pi,6pi]. 

My answer: 

I'm having an awful time with this one. I'm able to graph the basic equation, but I'm confused on how to incorporate the interval into it. Does it actually change the graph in any way? How would I do this one? 

 

2. Evaluate (5 + 5sqrt3i)^7 using DeMoivre's theorem. Write our your answer in rectangular form. 

My answer: 

r = sqrt(a^2 + b^2) and theta = tan^-1(b/a)  

R = sqrt (5)^2 + (5sqrt3)^2  

R = sqrt 25 + 75  

So r = 10  

Theta = tan^-1 (b/a)  

T = tan^-1 ((5sqrt3)/5) 

T = pi/3 or 60 degrees  

So (5 + 5sqrt3i)^7 is equal to 10cis(60degrees)  

[10cis(60)]^7  

= 10^7 cis(60)  

X = rcostheta  

Y = rsintheta  

X = 10^7 cos(60) = 10^7 * ½ = 5 * 10^6 

Y = 10^7 sin (60) = 10^7 * sqrt3/2 = 5sqrt3 * 10^6 

So for our final answer: 10^7(1 + I sqrt3)/2

 

3. A helicopter departs from an airport flying at a speed of 90 miles per hour at a bearing of 250° for 20 minutes before landing on a helipad. Determine the helipad’s location in relation to the airport. Round to the nearest tenth of a mile. Show all your work and explain your reasoning. 

My answer: 

1 hr = 60 min  

90/60 = 1.5 miles/min   

1.5 x 20 = 30  

So the distance the helicopter traveled is 30 miles. 

30 cos 250 degrees = -10.3 

30 sin 250 degrees = -28.2 

The helipad would be about 28.2 miles West and 10.3 miles South of the airport.

 May 6, 2021
 #1
avatar+781 
+1

Very thoughtful post, guest! 2&3 looks good to me but I do not know what sec is.

 #3
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Thank you so much!

Guest May 6, 2021
 #2
avatar+121055 
+2

1. Graph the function f(x) = -sec(x/4)+3 on the interval [-2pi,6pi]

 

The  " -"    flips  the  normal  secant   graph over  the x axis

 

The (x/4)   lengthens  the  normal period  by  a  factor  of  4

 

The " +3 "  shifts everything  up  by 3 units

 

Here's  the  normal graph   and  the  transformed graph on  [-2pi, 6pi]

 

https://www.desmos.com/calculator/ptwct2d3ec

 

 

cool cool cool

 May 6, 2021
edited by CPhill  May 6, 2021

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