Simplifying Expressions with Double-Angle Formulas

Date: 05/25/98 at 16:20:26
From: Stacey Pickett
Subject: Trigonometry

Express the following as a single function:
  
   1) 1 - 2*sin^2(4x)

   2) 2*sin(5x)*cos(5x)

Please help! I have no clue where to start. 

Thank you.

Date: 05/28/98 at 08:07:08
From: Doctor Mateo
Subject: Re: Trigonometry

Hello Stacey, 

To write either one of the expressions that you have stated as a 
single trigonometric function, you will need to understand the 
relation of the expression to the trigonometric identity that you are 
now studying in class.

What that means is this - you are probably studying the double-angle 
identities right now in class. 

The double-angle identities that you are studying relate to earlier 
identities in trigonometry as well, but the real question stems from 
your understanding of what the double-angle identities are saying:

   Identity: sin(2*A) = 2*sin(A)*cos(A)

The identity above says that we take an angle, which we call A, and 
double it. That is, when we multiply it by two and take the sine of 
it, it is equal to the same thing as multiplying the sine of angle A 
by the cosine of angle A and then multiplying the result by 2.

Let's look at an example. Suppose that you want to rewrite:
 
   2*sin(7w)*cos(7w) 

as a single trigonometric function.

Observe that 2*sin(7w)*cos(7w) looks like 2*sin(A)*cos(A). In this 
example, you can say that the angle is 7w.

Now the identity says that 2*sin(A)*cos(A) = sin(2*A), so if the angle 
is 7w, that means that if we want to use the identity we could say 
that A = 7w.

So  2*sin(7w)*cos(7w) = sin(2*7w) = sin(14w).
 
The other identity you appear to be studying right now is the double 
angle for cosine:

   Identity: cos(2*A) = 1 - 2*sin^2(A)

Again, the angle is represented by A. Let us look at another example.
Say that you want to rewrite as a single function:

   1 - 2*sin^2(0.25B) 

What does your identity say?  It says consider the measure of the 
angle first. The angle in this example is 0.25B, so 0.25B = A in the 
rule for this identity. 

Now let's use the identity. It says that 

         1 - sin^2(A) = cos(2*A), so that means 
     1 - sin^2(0.25B) = cos(2*0.25B)
                      = cos(0.5B)

Be careful to remember that cos(2*A) can come from other forms, too. 
For example, 

  cos^2(A) - sin^2(A) = cos(2*A)  and
       2*cos^2(A) - 1 = cos(2*A). 

The important thing to remember is that you work them all the same 
way. Ask yourself what the angle is, then what identity the problem 
most closely resembles.

-Doctor Mateo,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/