1. ## Double Identity Problems

I know it has something to do with the double identities, but I don't know where to start. Thanks in advance.

If x is a positive acute angle and $sin x = \frac{1}{2}$ , what is $sin 2x$ ?

2. Originally Posted by ThaBeretta
I know it has something to do with the double identities, but I don't know where to start. Thanks in advance.

If x is a positive acute angle and $sin x = \frac{1}{2}$ , what is $sin 2x$ ?
You need to know that sin(2x) = 2 sinx cos x (this is one of the famous double angle formulas).

You're given sin x so now all you have to do is calculate cos x. I assume this will be similar to problems you have previously done.

Then substitute both into the above formula.

3. Originally Posted by mr fantastic
You need to know that sin(2x) = 2 sinx cos x (this is one of the famous double angle formulas).

You're given sin x so now all you have to do is calculate cos x. I assume this will be similar to problems you have previously done.

Then substitute both into the above formula.
Hint for finding $\cos{x}$:

Remember another famous identity, $\sin^2{x} + \cos^2{x} = 1$.

4. Originally Posted by mr fantastic
You need to know that sin(2x) = 2 sinx cos x (this is one of the famous double angle formulas).

You're given sin x so now all you have to do is calculate cos x. I assume this will be similar to problems you have previously done.

Then substitute both into the above formula.
I assumed I had to use that identity because it's the only one that starts with $sin 2x$, but how would I calculate the value of $cos x$ if there are two unknowns:

sin2x = 2sinx cosx

sin2x = 2 (1/2) cos x

sin2x = cosx

5. Originally Posted by ThaBeretta
I assumed I had to use that identity because it's the only one that starts with $sin 2x$, but how would I calculate the value of $cos x$ if there are two unknowns:

sin2x = 2sinx cosx

sin2x = 2 (1/2) cos x

sin2x = cosx
You get the value of cosx from the known fact that sin x = 1/2 and x is in the first quadrant.

You then substitute the values of sin x and cos x into 2 sin x cos x to get the value of sin(2x).

6. Given that information, cos x must then equal radical 3/2, correct? So the final answer is radical 3/2 (sin x is 1/2 and that is multiplied by 2, giving you 1).

7. Originally Posted by ThaBeretta
Given that information, cos x must then equal radical 3/2, correct? So the final answer is radical 3/2 (sin x is 1/2 and that is multiplied by 2, giving you 1).
Yes.

Another way of doing it would be to note that if sin x = 1/2 and x lies in the first quadrant, then x = pi/6. Hence 2x = pi/3 and so sin(2x) = sin(pi/3).