Solving trig equations- Stuck at the double angle forumla

**coasterguy** · Feb 20th 2009, 06:59 PM

I find it hard to type the problem on the computer, so I included a picture with a note on where I am stuck. I know I can't divide "1" by "sin2theta" to get 30 degrees, so I don't know what to do next. Thanks.

**JoshHJ** · Feb 20th 2009, 07:15 PM

$ sin (2\theta) = 1 $
$ \arcsin (1) = 2\theta $
$ \frac {\arcsin (1)}{2} = \theta $

The sine function is 1 at 90 degrees or $\frac {\pi}{2}$
Therefore, $\theta = \frac {\pi}{2}$

**coasterguy** · Feb 20th 2009, 08:04 PM

Originally Posted by JoshHJ

$ sin (2\theta) = 1 $
$ \arcsin (1) = 2\theta $
$ \frac {\arcsin (1)}{2} = \theta $

The sine function is 1 at 90 degrees or $\frac {\pi}{2}$
Therefore, $\theta = \frac {\pi}{2}$

Thanks, but the answer I am given for the problem is: "45°". We haven't done arcsin yet.

**JoshHJ** · Feb 20th 2009, 08:22 PM

Oops, the answer is sin 45. I meant to type pi/4. Though, if you are doing trig identities you should have gone over arcsin, which is the inverse function of sine.

**Reckoner** · Feb 20th 2009, 08:24 PM

Originally Posted by coasterguy

I find it hard to type the problem on the computer, so I included a picture with a note on where I am stuck. I know I can't divide "1" by "sin2theta" to get 30 degrees, so I don't know what to do next. Thanks.

As you have written, $0^\circ\leq2\theta<720^\circ\text.$ Where, in the interval $0^\circ\leq\phi<720^\circ$ , is $\sin\phi$ equal to 1? That would be $\phi = 90^\circ, 450^\circ\text.$ But in our case, the angle is $\phi = 2\theta\text.$ So

$2\theta = 90^\circ\Rightarrow\theta = 45^\circ$

and

$2\theta = 450^\circ\Rightarrow\theta = 225^\circ\text.$

If you substitute these back into the original equation, you'll see that the solution is $45^\circ,$ and the second angle of $225^\circ$ is an extraneous solution introduced when you squared both sides.

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