# Half angle or double angle Identities

• Jul 21st 2009, 01:36 PM
hellojellojw
Half angle or double angle Identities
I don't understand them at all.
The problem is to evaluate tan(22.5) using exact values.
How should I do this?
• Jul 21st 2009, 01:47 PM
Krizalid
$\tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.$

Actually is $\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}},$ but the angle is positive, hence our answer is positive, and that's preserves the positive sign.
• Jul 21st 2009, 02:10 PM
hellojellojw
Quote:

Originally Posted by Krizalid
$\tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.$

Actually is $\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}},$ but the angle is positive, hence our answer is positive, and that's preserves the positive sign.

hmm can I assume tan (22.5) is equal to tan 1/2 x?
• Jul 21st 2009, 02:52 PM
pickslides
Krizalid is suggesting $x = \frac{\pi}{4}$
• Jul 21st 2009, 06:20 PM
hellojellojw
Well, that was easier than I expected. Thanks.