30)+Derive+sum+and+difference+and+double+angle+IDs

= = Link to Instructional Video: [] = = = Why I Chose This Topic = This section was one of my favorites, although it took me a while to memorize each identity and how we derived them. Once we derived each identity, I was able to easily understand how to use them within the problems we were given, although it took some erasing. Even though we were told we had to derive the difference ID for cosine first, it was not something that came to any of us easily. Using substitution and simplifying has always been one of the concepts in math that came easily to me, so once I got started, it was simple to come out with the ending identities for each. Using them correctly was the more difficult part. To derive the sum and difference ID for cosine took a large amount of thinking to figure out which formulas to use. Even though the process for the difference ID for cosine was so long, once we knew it, it was much easier to derive the others. We had to use previously learned formulas and identities to derive the sum and difference ID for both cosine and sine. Once we knew how to find the sum and difference identities, it was much easier to derive the double angles identity. We used past identities to come up with the new one.

Explanation of Deriving the Sum and Difference Identities
The sum and difference identities are used when you are asked to find the cosine, sine, or tangent of angle that is not found on the unit circle or the addition/subtraction of two angles. For all three trig functions, the problem will be set up like Cos(α+ β) where α and β represent two different angles. Once you derive the sum and difference identity for cosine, you can easily derive the sum and difference identity for sine. The sum and difference identity for tangent is derived different, but still is based on previously learned identities.

Explanation of Deriving the Double Angle Identities
Deriving the double angle identities is much easier. To derive the sine double angle identity, you must start out with sin(2x). sin(2x) is equal to sin(x+x). Now we can use the sum identity for sine. Input the two angles, "x" into the identity, like this: sinxcosx+cosxsinx. This can now be simplified to 2sinxcosx, which is the double angle identity for sine. sin(2x) = 2sinxcosx

To derive the cosine double angle identity, start out like you did with sine, cos(2x) = cos(x+x). Now substitute the two angles of "x" into the sum identity for cosine, like this: cosxcosx-sinxsinx. This can be simplified to (cosx)^2 - (sinx)^2. This is the double angle identity for cosine. cos(2x) = (cosx)^2 - (sinx)^2

Deriving the tangent double angle identity is very similar. You start out with tan(2x) = tan(x+x). Substitute the angles, like you did for the last two, into the tangent sum identity, like this: tanx+tanx/ 1-tanxtanx. This can then be simplified to 2tanx/ 1-(tanx)^2 and this is the tangent double angle identity. tan(2x) = 2tanx/ 1-(tanx)^2

= Example 1: Easy - Derive the Double Angle Identity for Sine =

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= Example 2: Hard - Derive the Tangent Sum Identity =



= Problem Set: =