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/
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