Ian G.

asked • 01/03/16

Maths Trig identities

Exact value of Sin pi/12
 
I got this answer to be  (√6 - √2 )/4 (part a)
 
I then done the half angled identity to be  √(2 - √3)/2 (part b)
 
Both these expressions are equivalent and I need to prove it so the question is 
 
By squaring the exact values obtained for sin(π/12) in parts (a)
and (b) (or otherwise), show that the two expressions are equivalent.
 
Really don't understand this any help would be great.

Mark M.

Since you are familiar with the formulas, I suggest you try the difference formula using:
 
sin (π/3 - π/4)      [π/12 = 4π/12 - 3π/12]
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01/03/16

Ian G.

The questions says I have to use the answers that I obtained in parts (a) and (b) and by squaring both expressions show that they are equivalent. I'm not following what you are saying.
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01/04/16

1 Expert Answer

By:

Ian G.

I can follow some of this and its very helpful so thank you so much.
 
I am struggling to follow your half angle identity the working I done was
 
sin[pi/12] = 1/2* (1 - cos(pi/6)
 
                 = 1/2 *(1 - √3/2)
 
                 = 1/2* (2/2 - √3/2)
  
                 = (2 - √3)/4
         
                 =√(2-√3)/4
 
                 = √(2-√3)/2
 
Is this the same as your answer but fraction been rationalised? 
 
 
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01/04/16

Eric C.

tutor
Yes sir. Sorry for the confusing typing; the formatting here makes it tough to make things clear and concise.
 
I'll take it more step-by-step with my method.
 
The half-angle formula for sine is as follows:
 
sin(Θ/2) = √((1 - cos(Θ))/2)
 
Remember that the square root encapsulates the entire term, including the 2 in the denominator.
 
Since we're evaluating sin(pi/12), we want Θ = pi/6
 
cos(pi/6) = √(3)/2
 
If we look at the term inside the square root, it will be:
 
1/2*(1 - √(3)/2)
 
= (1/2 - √(3)/4)
 
This is the term inside the square root, so sin(pi/12) =
 
√(1/2 - √(3)/4)
 
Squaring this term, you get what's inside the expression.
 
1/2 - √(3)/4
 
Now let's square your term and see what happens.
 
(√(2-√3)/2)2
 
= (2 - √(3))/4
= 2/4 - √(3)/4
 
= 1/2 - √(3)/4
 
So we got the same result, just with a different approach.
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01/04/16

Ian G.

Thanks I got there in the end and thanks to you. I would never have thought about foil and that in this question was nagging my brain so now onto the next :)
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01/05/16

Eric C.

tutor
Completely understandable! We all recognize that
 
(x - y)2 = x2 - 2xy + y2
 
but when asked to square something like
 
(√(6) - √(2))2
 
our brains just freeze.
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01/05/16

Ian G.

I posted another question to do with vectors if you are any good with them ?
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01/05/16

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