Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same.
5 sin6(x)
To rewrite the expression in terms of the first power of the cosine, we can use the identity sin^2(x) + cos^2(x) = 1.
First, let's recall the identity sin(2θ) = 2sin(θ)cos(θ). Using this identity, we rewrite the expression 5 sin(6x) as follows:
5 sin(6x) = (5/2)(2 sin(6x))
Next, we'll use the identity sin(1/2θ) = √[(1 - cos(θ))/2] to rewrite sin(6x) as a function of the first power of cosine:
sin(6x) = 2 sin(3x) cos(3x) = 2(√[(1 - cos(6x))/2])(cos(3x))
Multiplying these two expressions together, we get:
(5/2)(2 sin(6x)) = 5 √[(1 - cos(6x))/2] cos(3x)
To graph both expressions and verify that they are the same, we can use a graphing utility. Here's a step-by-step guide on how to graph the expressions using Desmos (a popular online graphing tool):
1. Go to the Desmos website (https://www.desmos.com/).
2. Click on "Graphing Calculator" in the top menu.
3. In the input bar at the top, enter the expression "5 sin(6x)".
4. Press "Enter" or click on the "+" button to graph the expression.
5. To graph the rewritten expression, enter the expression "5 √[(1 - cos(6x))/2] cos(3x)" in the input bar.
6. Press "Enter" or click on the "+" button to graph the rewritten expression.
Now you should see the graphs of both expressions. If they are identical, it means that the rewritten expression is indeed equivalent to the original expression in terms of the first power of the cosine.