A trig question -- sine with half angle identity vs. sin (A-B)

Question

In trying to find the value of sin 165 degrees, I get the answer (sq rt (2 - sq rt 3))/2 using 1/2 angle equality -- 30 degrees / 2 -- and (sq rt 6 - sq rt 2)/4 using 180 deg - 15 deg. Calculator says they = each other. How can I convert one value to the other?  (Because 165 deg is in Quad II, sine is positive and reference angle is 15 deg)

Answer 1

Steve,

 

The direct answer to your question is "yes, there's a way to convert one value to the other."  It's a little bit off the beaten path, and I will admit that I'm not an expert on the exact technique, so this might not be the most efficient approach.  But, it worked, so I figured I'd share, and others can refine the answer if there's a better way...

 

First, let's make sure we have the correct numbers for the result you got with the half-angle formula for sin(15).  I get:

 

√((1-√3/2)/2)

 

So, the whole thing is under the square root, including the division by 2.  So, if we split up the 2, it looks like this: 

 

√(1/2 - √3/4)

 

The trick I know of for "de-nesting" a square root is to get the radical in this form:

 

√(a±2√b)

 

That doesn't quite look like what we have, but if you can write the expression under the radical as a±2√b, then there's a way to de-nest it. 

 

We have:

√(1/2 - √3/4)

 

We need the coefficient of the √3 to be 2. So, we factor 1/8 out of 1/4, leaving 2. If you do the same thing to the 1/2 and factor out 1/8, that leaves 4. Sooooo, we've got:

√[(1/8)(4-2√3)] = √(1/8)*√(4-2√3)

 

We can leave the 1/8 thing for later, but now, we have the nested radical in the right pattern. 

 

Ready for the trick?

 

- Find two numbers x & y such that they add up to "a" and multiply to "b."

- Make x the bigger of the two numbers.

- √(a-2√b) can be reduced to √x-√y.

 

In our problem, we've got √(4-2√3), so "a" is 4, and "b" is 3.

 

- x & y need to add to 4, and multiply to 3.  3 and 1 will do it.

- Make x the bigger of the two numbers.  So, that's 3, and y is 1.

- √(4-2√3) is therefore equal to √x-√y, or √3 - √1. √1 is 1, so we get √3 - 1.

 

We still have to deal with the part that we factored out, which was √(1/8), which I think ends up being √2/4 after you rationalize it?  So, the final answer is:

 

 √2/4 * (√3 - 1)

 

Re-distribute the √2 to the expression inside the parentheses, and poof, I think that matches the answer you'd get using sin(180-15).

 

Hope this helps...

 

 

 

Answer 2

sin(165)=sin(180-15)=sin(180)cos(15)-cos(180)sin(15)

                               =(0)cos(15)-(-1)sin(15)

                               =0+sin(15)

                               =sin(15)

                               =0.2588190451