Trig+Identities+9-30

At the beginning of class we talked about some very important administrative topics:
 * Wiki commenting should be done within 48 hours after the page is posted.
 * All revisions must be completed before the end of the quarter (ending in 2 weeks). However, it is good to keep in mind that there is no school Monday the 10th and Wednesday the 12th we are taking the PSAT.
 * We discussed the possible days for the date of our next test, and wrote our preference on a sheet of paper.

__**Double Angle Formulas:**__ There are multiple ways to write the double angle formulas, but the default forms are:

math $ \sin 2u=2\sin u\cos u $ math

math $ \cos 2u=\cos ^{2}u-\sin ^{2}u $ math

There are alternate ways to write each of these formulas which are not as important to memorize, but could be useful to recognize when proving other trig identities. Here are the alternate forms and how to get them from the original default formulas.

One alternate form for cos(2u) is: math $ \cos 2u=\cos ^{2}u-\sin ^{2}u $ math math $ \cos 2u=1-\sin ^{2}u-\sin ^{^{2}}u $ math math $ \cos 2u=1-2\sin ^{^{2}}u $ math

Another alternate form for cos(2u) is: math $ \cos 2u=\cos ^{2}u-\sin ^{2}u $ math math $ \cos 2u=\cos ^{2}u-(1-\cos ^{2}u) $ math math $ \cos 2u=2\cos ^{2}u-1 $ math

It is not really necessary to memorize these alternate forms, but simply know how to get from the default forms to them. These double angle formulas are also useful for memorizing Angle Sum Identities as well. For example, to memorize **sin(x+y)** is to remember that it is virtually the same as sin(u+u)= sin(2u). Therefore, just remember that there must be two pairs of sin(u)cos(u) to get the 2 in the double angle identity. Then you just have to substitute x and y for the u's and you have the Angle Sum Identity. Likewise, for cos(x+y) you just have to remember there must be cos(u)sin(u)-sin(u)sin(u) to get the squares in the double angle formula for cos(2u). Again, all you have to do is substitute x and y for the u's.

__**Examples of Applications of the Double Angle Formulas:**__ __ Example 1: (__Working with the left side first and changing it until it equals the right is easiest with proving this identity) math $ \sin 2x=(\tan x)(1+\cos 2x) $ math math $ \tan x(1+\cos 2x)= \tan x(1+\cos ^{^{2}}x-\sin ^{2}x) $ math math $ =\tan x(1+cos^{2}x-(1-cos^{2}x)) $ math math $ =\tan x(2cos^{2}x) $ math math $ =\frac{\sin x}{\cos x}\cdot 2\cos ^{2}x $ math math $ =2\sin x\cos x $ math math $ =\sin 2x $ math

__Example 2:__ (Working each side separately and then having them meet in the middle is easiest for this one) math $ \frac{\cos 2u}{1-\sin 2u}=\frac{1+\tan u}{1-\tan u} $ math On the left side: math $ \frac{\cos 2u}{1-\sin 2u}=\frac{\cos ^{2}u-\sin ^{2}u}{1-\sin x\cos x} $ math On the right side: math $ \frac{1+\tan u}{1-\tan u}=\frac{\frac{\cos u+\sin u}{\cos u}}{\frac{\cos u-\sin u}{\cos u}} $ math math $ =\frac{\cos u+\sin u}{\cos u-\sin u}\cdot \frac{\cos u-\sin u}{\cos u-\sin u} $ math math $ =\frac{\cos ^{2}u-\sin ^{2}u}{\cos ^{2}u-2\cos u\sin u+\sin ^{2}u} $ math math $ =\frac{\cos^{^{2}}u-\sin ^{2}u}{\cos ^{2}u+\sin ^{2}u-2\cos u\sin u} $ math math $ =\frac{\cos ^{2}u-\sin ^{2}u}{1-2\sin u\cos u} $ math math $ =\frac{\cos 2u}{1-\sin 2u} $ math

__**Double Angle Formulas ---> Half Angle Formulas**__ For the sake of example we used one of the simplest double angle formulas: math $ \cos (2\Theta) =1-\sin ^{2}\Theta $ math Let: math $ x=2\Theta ; \Theta =\frac{x}{2} $ math math $ \cos x=\sin ^{2}(\frac{x}{2}) $ math math $ \cos x-1=-2\sin ^{2}(\frac{x}{2}) $ math math $ 1-\cos x=2\sin ^{2}(\frac{x}{2}) $ math math $ \frac{1-\cos x}{2}=\sin ^{2}(\frac{x}{2}) $ math math $ \pm \sqrt{\frac{1-\cos x}{\sin (\frac{x}{2})}} $ math

It is important to note that there was multiplication across the equals sign in this proof, which is usually not allowed. However, it only doesn't work if it will be 0, so multiplying by 2 or 1/2 is allowed and can be extremely helpful in problems such as this one.