Trig Identities

Question

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.

Answer 1

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

 

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.)

 

 

Answer 2

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. 

 

Answer 3

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

Answer 4

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=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)

cos(-30)=adjacent/hypotenuse=-sqrt(3)/2

sec=1/cos

sec=1/[-sqrt(3)/2]=-2/sqrt(3)

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