Use the addition formulas for sine and cosine to simplify each expression. cos((π)/(3)-θ)-cos((π

Use the addition formulas for sine and cosine to simplify...

Key Concepts

Simplification of Trigonometric Expressions

This concept involves applying identities like the addition and subtraction formulas to rewrite trigonometric expressions in a more compact or useful form. It often includes combining like terms, factoring common factors, and using identities to reduce complex expressions, which is essential in solving equations and computing values in trigonometry.

Trigonometric Addition and Subtraction Formulas

These formulas allow the expression of the sine or cosine of a sum or difference of angles in terms of the sines and cosines of the individual angles. For instance, the cosine addition formula is cos(A + B) = cos A cos B - sin A sin B, and the cosine subtraction formula is cos(A - B) = cos A cos B + sin A sin B. They are fundamental tools for simplifying and transforming trigonometric expressions.

Transcript

00:01 Hello everybody.

00:02 In this video, i'm going to be showing you how to solve exercise 23 in chapter 9, section 1 of cohen's precalculus 7th edition.

00:11 Now, in this problem, they want us to use the addition formulas for sine and cosine to simplify the expression, cosine of pi over 3 minus theta, minus cosine of pi over 3 plus theta, where theta is some angle.

00:26 Now, to do this, what we first want to do is recall the addition formulas for cosine, which we can express in this condensed form.

00:33 For two angles, s and t, the cosine of s plus or minus t, is equal to the cosine of s times the cosine of t, minus plus the sign of s times the sign of t.

01:00 And this really represents both equations for cosine, but this plus or minus and minus plus just indicate that a different version of this equation must be used, depending on whether or not the terms and cosine's argument are subjection.

01:12 From each other were added together.

01:16 And so with this formula, all we need to do is plug both of these terms into this expression and expand, using the substitutions s equals pi over 3 as well as t equals theta.

01:32 So let's do this down here.

01:35 For the first term, cosine of pi over 3 minus theta, we want to use the minus version of this equation, which means that we use a plus over here.

01:43 We have that this is equal to the cosine of s, which is the one, is pi over 3 times the cosine of t, which is theta, plus the sign of s, which is pi over 3, times the sign of t, which is theta.

02:05 Now subtracted from this, we have the expansion for the second term...

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