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Explain how you could determine the exact value of (sec 7pi/6) if you know the value of (sin 11pi/6).
Prove with diagrams.

I've no idea how to do this question that's worth 10 marks.
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4 Answers

Hi Lawson,
Unfortunately, I am not sure how to draw diagrams here so I will try hard to explain the steps well. 
Given the information we have already:
1) (sin 11pi/6) is the same as (sin pi/6) in the fourth quadrant of an x, y axis graph. Therefore,
(sin 11pi/6) = (-sin pi/6) which we know the value of.
2) (sec 7pi/6) is the same as (sec pi/6) in the third quadrant of an x,y axis graph. Therefore,
(sec 7pi/6) = (-sec pi/6)
3) sec = 1/cos ==> (-sec pi/6) = (-1/cos(pi/6))
  • Using SOHCAHTOA (Sine = Opposite/Hypotenuse), we can then draw a right-angled triangle with one angle being pi/6 and the length of the side opposite the angle being the numerator of the value of (-sin pi/6), and the length of the hypotenuse being the denominator of the value of (-sin pi/6).
  • Since we have the length of one side of the right-triangle and the length of the hypotenuse, we can find the length of the third side by using the Pythagorean Theorem:
                      (side 1)2 + (side 2)2 = (hypotenuse)2
  • From step 3 we know that (-sec pi/6) = (-1/cos(pi/6)). We have already formed a right angled triangle with one angle being pi/6 and we know the length of all three sides. Therefore again using SOHCAHTOA (Cosine = Adjacent/Hypotenuse => 1/cosine = Hypotenuse/Adjacent), you can now find the exact value of (-sec pi/6). 
Sorry about all the writing but I really hope this makes sense. Let me know if you have any questions. 
Let's use degrees instead of radians:
7pi/6 = 210° and 11pi/6 = 330°
Draw an x-y coordinate system and draw two lines at 210° and 330°, i.e., in the 3rd and 4th quadrant.
Draw two lines parallel to the y-axis such that you will get two congruent 30-90-60 triangles. They will have the same three sides (opposite, adjacent, hypotenuse). 
Now remember:
sec x = hypotenuse/adjacent 
sin x = opposite/hypotenuse
Combine them:
sec x sin x = opposite/adjacent = tan x
so that
sec 30° sin 30° = tan 30° = 1/√3
sec 210° = -sec 30° = -(1/√3) / sin 30 = -(1/√3)/(-sin 330) = -2/√3
(The minus signs come from the fact that the two adjacent sides have opposite signs.)
cos 7pi/6 = -cos pi/6 = -sqrt(3)/2, using reference angle
sec 7pi/6 = 1/cos 7pi/6 = -2/sqrt(3) = -2sqrt(3)/3 <==Answer
180 degrees= pi radians
sin 11pi/6=sin 11*180/6=sin 11 30 degrees=sin 330 degrees=-sin 30 degrees=-1/2
-sin 30 is in the fourth quadrant
draw a right triangle with 30 degrees at the vertex and one leg part of the positive x-axis and the other leg perpendicular to the positive x-axis
sin(-30)=-1/2 so the perpendicular side is -1 and the hypotenuse is 2
therefore the other leg is, from the Pythagorean Theorem a^2+b^2=c^2, a^2+1=4 and a^2=3 giving us
we got the answer from the value of sin 11pi/6
however we could have done the following:
sec 7pi/6=sec 7*30=sec 210=-sec 30=-2sqrt(3)/3=-2/sqrt(3)  [multiply both terms by sqrt(3)]
-sec 30 is in the third quadrant going clockwise