Question 1205998
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Refer to the diagram made at this link
<a href="https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1205915.html">https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1205915.html</a>
(credit goes to tutor Edwin McCravy)


The plane wants to fly directly west along the longer blue vector.
The wind pushes the plane along the diagonal blue vector such that the plane's actual path is along the red vector. 
It's like a game of tug-of-war.


Let's label the points of the triangle A,B,C where I'll start in the top left corner and work clockwise.
The plane starts at B. It wants to move to A. Instead the wind pushes it to C.


If we let D represent a point directly south of point A, then angle CAD is the S30°W angle mentioned. 
It's where you look directly south and then turn 30° westward. 
Add on 90 degrees (the measure of angle DAB) to determine the measure of angle CAB.


So,
angle CAB = angleCAD+angleDAB = 30+90 = 120
In short,
Angle CAB = 120°


Use the law of cosines as tutor Edwin mentions to determine that {{{R = sqrt(200^2+25^2-2(200)(25)cos(120^o)) = 213.6000936329}}} approximately. 
This rounds to <font color=red>213.6 mph</font>.
This is the groundspeed along the red vector. The original speed (200 mph) has been increased to roughly 213.6 mph due to the wind pushing it slightly.


Then we use the law of sines to find the smallest angle.
The smallest angle is opposite the smallest side (AC = 25)
sin(A)/a = sin(B)/b
sin(120)/213.60009 = sin(B)/25
sin(B) = 25*sin(120)/213.60009
sin(B) = 0.10136
B = arcsin(0.10136)
B = 5.81749
B = 5.82
The <font color=red>smallest angle is roughly 5.82°</font> which is opposite side AC = 25.
This is the approximate measure of angle ABC. The angle complementary to this is 90-5.82 = 84.18°


If we let point E be directly south of B, then angle EBC = 84.18°
The plane is traveling along the red vector in the direction of <font color=red>S84.18°W</font>
Place yourself at point B, face directly south, and then rotate roughly 84.18° westward so that you're aiming for point C.


The largest angle is 120° we found earlier (angle CAB).
The <font color=red>remaining angle is</font> 180-120-5.82 = <font color=red>54.18° approximately.</font>
This is angle ACB.


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Summary:<ol><li>R = <font color=red>213.6</font></li><li>smallest angle = <font color=red>5.82°</font></li><li>largest angle = <font color=red>120°</font></li><li>remaining angle = <font color=red>54.18°</font></li><li>direction of the plane's new path = <font color=red>S84.18°W</font></li></ol>
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