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## Trig Identities

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.

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.

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:

sin x = opposite/hypotenuse

Combine them:

sec x sin x = opposite/adjacent = tan x

so that

sec 30° sin 30° = tan 30° = 1/√3

Therefore,

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
So,
sec 7pi/6 = 1/cos 7pi/6 = -2/sqrt(3) = -2sqrt(3)/3 <==Answer
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=opposite/hypotenuse
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
a=sqrt(3)