Algebra/Trig Review - Pauls Online Math Notes - Lamar University

Algebra/Trig Review2x = 1 ( + ( x))cos ( ) 1 cos 22This is really the second formula from Problem 4 in this section rearranged and isVERY useful for eliminating even powers of cosines. For example,2 ⎛1⎞5cos ( 3x) = 5⎜( 1+cos( 2( 3x)))⎟⎝2⎠5= ( 1 + cos ( 6 x))2Note that you probably saw this formula written as⎛ x ⎞ 1cos⎜⎟ =± 1 + cos⎝2⎠2in a trig class and called a half-angle formula.( ( x))6.2sin ( x ) =(In terms of cosine to the first power)Solution2x = 1 ( − ( x))sin ( ) 1 cos 22As with the previous problem this is really the third formula from Problem 4 in thissection rearranged and is very useful for eliminating even powers of sine. Forexample,( )( t) = ( t)4 24sin 2 4 sin 2⎛1⎞= 4⎜( 1−cos( 4t))⎟⎝2⎠⎛1⎞2= 4⎜⎟( 1− 2cos( 4t) + cos ( 4t))⎝4⎠1= 1− 2cos( 4t) + ( 1+cos( 8t))23 1= − 2cos( 4t) + cos( 8t)2 2As shown in this example you may have to use both formulas and more than once ifthe power is larger than 2 and the answer will often have multiple cosines withdifferent arguments.Again, in a trig class this was probably called half-angle formula and written as,⎛ x ⎞ 1sin ⎜ ⎟ =± ( 1 − cos( x))⎝2⎠222© 2006 Paul Dawkins 61http://tutorial.math.lamar.edu/terms.aspx