Expert: Paul Klarreich - 3/4/2011

Question
Hello, I have a couple questions.

1. Find the exact value of sin(x/2) when secu=(-7/2)
I know you have to use one of the half angle formulas, my answer doesn't make sense. The formula is sin(x/2)=squareroot((1-cosx)/(2)) but I've tried it several times and I keep getting odd numbers that don't make any sense.

2. Rewrite the expression i terms of the first power of the cosine (in simplest form): sin^4xcos^2x.
Here, I confess that I'm not really sure what the question is asking. What makes the most sense is to use the Power Reducing Formula, but I'm not sure how to break down the expression. Would it be something like sin^2xsin^2xsin^x?  And then you would plug in the power reducing formula for all three terms?

Thank you very much for any help you can give,
Meghan.

Answer
Questioner: Meghan
Country: United States
Category: Calculus
Private: No
Subject: precalculus question
Question: Hello, I have a couple questions.

1. Find the exact value of sin(x/2) when secu=(-7/2)
I know you have to use one of the half angle formulas, my answer doesn't make sense. The formula is sin(x/2)=squareroot((1-cosx)/(2)) but I've tried it several times and I keep getting odd numbers that don't make any sense.

I assume you meant to write:  sec x = -7/2.

Then  cos x = -2/7 and you can just plug that in.  BTW, 'exact value' means including square roots.
........................................

2. Rewrite the expression in terms of the first power of the cosine (in simplest form): sin^4xcos^2x.
Here, I confess that I'm not really sure what the question is asking.

>> I think that means an expression with  cos(...) in it, and no powers.

What makes the most sense is to use the Power Reducing Formula,

>> Never heard of it.  PLEASE don't assume I have your textbook.

but I'm not sure how to break down

>> mathematicians don't break down things.  Maybe firemen break down doors to get into a building, etc..

the expression. Would it be something like sin^2xsin^2xsin^x?  And then you would plug in the power reducing formula for all three terms?

Now, then:

You could use the 'half-angle' trick, most commonly used in calculus, to get rid of even powers of sin x or cos x:
          1 - cos 2x
sin^2(x) = ----------
              2

          [ 1 - cos 2x ]^2
sin^4(x) = [----------  ]
          [    2       ]

           1 - 2 cos 2x + cos^2(2x)
sin^4(x) = -----------------------
                     4

           1 - 2 cos 2x + (1 + cos(4x)/2
sin^4(x) = ------------------------------
                     4

           2 - 4 cos 2x + 1 + cos(4x)
sin^4(x) = ---------------------------
                     8

           3 - 4 cos(2x) + cos(4x)
sin^4(x) = -----------------------
                     8


          1 + cos(2x)
cos^2(x) = ----------
              2

Multiply those.  Right now I don't have anything better for you.

Thank you very much for any help you can give,
Meghan.