Given cos(α)=−8/9 and α is in quadrant III, find the exact value of cos(α/2).

Answer 1

This question requires that you know how to manipulate trigonometric functions with algebra.

Please let x = alpha (so it's easier for me to show this. It doesn't change the value)

We can only manipulate x when it's isolated so we do that using inverse cos or arccos

cos(x) = -8/9 Our base set up (given)

x = arccos(-8/9) Take the arccos of both sides. This cancels cos

x/2 = arccos(-8/9)/2 Divide both sides by 2 because we can now manipulate our x

Now is when I'm sure you ran into some issues. arccos(-8/9) has TWO answers but most calculators will only tell you one of them. It just so happens that the one they tell you is not the one you need. But you can use the one they give you to determine the one you need.

arccos(-8/9) gives .8485 π radians when put into the calculator. This and the one we are looking for are equidistant from π radians which is arccos(-1). So what we can say is that arccos(-8/9) = [1 + (1 -.8485)] π radians where (1-.8485) π radians is the distance from π radians.

Now we know that our arccos(-8/9) = [1 + (1 -.8485)] π radians for our problem.

Substituting in for our equation we have that x/2 = ([1 + (1 -.8485)] π radians )/2 Now, the fun part, we get to take the cos of both sides once again to get the exact value that it is asking for.

cos(x/2) = cos[ ( [ 1 + ( 1 -.8485 ) ] π radians ) /2 ]

cos(x/2) = [cos( arccos(-1) + [ arccos(-1) - arccos(-8/9) ] ) ] / 2 Use this in any calculator and it should get you the correct answer

I used π radians because it's nicer to look at. Your question is likely expecting different units, so just follow this algorithm for those units and you should be able to get the correct answer.

It's also a half-angle identity so you could just do that but that isn't as fun.

cos (α/2) = ±√(((1 + cos(α)) )/ 2) gives cos α/2 = -√(((1 - 8/9) )/ 2) we choose '-' because it's in Quadrant III

Answer 2

cos α/2 = -((√(1 + 8/9) / 2)