Use the addition formulas for sine and cosine to simplify each expression. cos(θ-(π)/(4))+cos(θ+

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

Key Concepts

Trigonometric Addition Formulas

These formulas express the sine or cosine of a sum or difference of angles in terms of the product of sine and cosine of the individual angles. They are fundamental in rewriting and simplifying trigonometric expressions by breaking them down into more manageable parts.

Sum-to-Product Identities

These identities convert sums or differences of trigonometric functions into products, which can assist in simplifying expressions or solving equations. They are particularly useful when the angles involved are symmetrically related, as they allow for a concise factorization of the expression.

Even Function Property of Cosine

The cosine function is even, meaning that cosine of a negative angle is equal to the cosine of the positive angle. This property is useful in simplification processes, as it permits the removal of negative signs inside cosine functions, thereby streamlining the expression.

Transcript

00:01 Hello, everybody.

00:03 In this video, i'm going to be showing you how to solve exercise 24 in chapter 9, section 1 of cohen's pre -calculus 7th edition.

00:11 Now, in this problem, they want us to simplify the expression, cosine of theta minus pi over 4, plus cosine of theta plus pi over 4.

00:22 Now, the first thing we want to do is recall the addition formulas for cosine, which can be condensed into the following equation.

00:29 For two angles, s and t, the cosine of s plus s.

00:34 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.

00:53 So all we need to do is take the two terms in our expression and plug them into this equation with the substitution s equals theta and t equals pi over four.

01:07 And then we just need to discern which version of the equation to use, the plus version or the minus version, depending on whether or not for each term, the arguments within the cosine are subtracted from each other or added to each other.

01:20 Let's start with the first term, cosine of theta minus pi over four.

01:25 Because these terms are subtracted from each other, we want to use the minus version of its equation, which means that we have a plus over here in the expansion.

01:34 So this gives us the cosine of s, which is theta, times the cosine of t, which is pi over four, plus the sine of s, which is theta, times the sine of t, which is pi over four.

01:55 And now added to this, we have the expansion of the second term.

01:59 And because its terms are added together, we're going to use the plus version of this equation, which means we have a minus over here in the expansion...

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