Hi Zekiguy. Your answers match mine, and I can follow your work, so I'm not sure what your instructor might be thinking.
When I work an exercise containing both radian and degree measure, I always write degree symbols on numbers representing degree measure. Could it be that your instructor would like to see degree symbols on your measures of 180º and 360º?
Are you using formulas as shown in class? Your work shows formulas for degree measure; I wonder if your instructor realized that you were evaluating cos
-1(2.5/3) with a device set to degree mode. In other words, some people might see the expression cos
-1(2.5/3) and mistakenly assume it always means radian measure. You could replace the angle expression cos
-1(2.5/3) with 33.5573º. (I'm really guessing, here.)
With your approach, I would also have used the Pythagorean Theorem to write (√11)/2 instead of 2.5∙tan(cos
-1(2.5/3)) -- for the height of triangle ABC. (That's just my preference; it doesn't change the answer.)
In my approach, I would have worked with radian measure throughout, using formulas from
this page. For finding the area of region R, I would not have subtracted the area of a triangle from the area of a sector (doubling the result). I would have used the formula for the area of a circular
segment (The area of region R is the same as the area of the segment with the angle doubled.)
?
