Rewrite Using the Product-to-Sum Identities (MathAngel369)

Описание: Using the Product-to-Sum Identities to Rewrite Trig Expressions
 
 
 
In this discussion, we are going to learn how to use the product-to-sum identities to rewrite trig expressions.
 
I have already proved the product-to-sum identities in a previous discussion, so check it out if you are interested (suggested videos are linked below)!

In our example, we have

cos(x)sin(3x)

Let us rewrite the above trig expression using one of our product-to-sum identities.

We begin with what we already know. Recall the product-to-sum formulas:

sin(A)cos(B) = 1/2 [sin(A+B) + sin(A-B)]

cos(A)sin(B) = 1/2[sin(A+B) -sin(A-B)]

cos(A)cos(B) = 1/2[cos(A+B) + cos(A-B)]

sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]

Since we have cos(x)sin(3x), we will use equation 2.

Thus, we have:

cos(x)sin(3x)

1/2[sin(x+3x) -sin(x-3x)]

Simplifying, we have:

1/2[sin(4x) - sin(-2x)]

By the odd identity, we can take the negative sign inside of the sine function, and pull it out. Thus, we have:

1/2[sin(4x) + sin(2x)]

You could distribute the 1/2, but I will leave it as a factor; either way is correct.

We have expressed a product of cosine and sine as a sum of cosine and sine.

The purpose of rewriting a product of cosine and sine into a sum is because, in calculus, if you are asked to evaluate the integral of a product of sine and cosine, one of the ways we can do so is by rewriting the product of sine and cosine as a sum.

However, for a trigonometry class, all you need to know is how to rewrite trig expressions from a product of cosine and sine into a sum of cosine and sine.

And that is how to use the product-to-sum identities to rewrite trig expressions.


Thank you for reading/watching!

I hope that I can assist you on your math journey!
 
Sincerely,
 
➕MathAngel369➕


 🎥 Suggested Video and Playlist:

Video: Proof of the Product-to-Sum Formulas for Cosine
   • Proving the Sum-to-Product Formulas for Co...  

Video: Proof of the Sum-to-Product Formulas for Sine
   • Proof of the Sum-to-Product Formulas for S...  

Playlist: Proving/Verifying and Using Trig Identities
   • Proving/Verifying and Using Trig Identities  






📝Blog Post:

Read my blog post associated with this video:

https://mathangel369.com/2022/05/15/h...




🔑 Math Course:

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You can preview about 10 minutes of the course for free before purchasing.

Proving/Verifying and Using Trig Identities

https://www.udemy.com/course/proving-...

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