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

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

Key Concepts

Angle Sum and Difference Identities

These identities express the sine or cosine of a sum or difference of two angles in terms of the sine and cosine of the individual angles. They are fundamental tools in trigonometry, enabling the transformation of expressions into forms that can be more easily simplified or analyzed.

Trigonometric Expression Simplification

This concept involves manipulating and reducing complex trigonometric expressions through techniques such as combining like terms, factoring, and applying trigonometric identities. It is essential for solving equations, evaluating expressions, and proving other mathematical results by revealing simpler underlying structures.

Transcript

00:01 Hello everybody.

00:03 In this video, i'm going to be showing you how to solve exercise 21 in chapter 9, section 1 of cohen's precalculus 7th edition.

00:12 Now, in this problem, they want us to use the addition formulas for sine and cosine to simplify the expression, sine of pi over 4 plus s, minus sine of pi over 4 minus s, where s is some angle.

00:26 Now, to do this, what we want to recall is both addition formulas for sign, which can be condensed in the following way.

00:34 The two angles, a and b, the sign of a plus or minus b, is equal to the sign of a times the cosine of b, plus or minus, the cosine of a times the sign of b.

00:55 And again, this represents two equations for the two different cases of whether or not the arguments within sign are added or subtracted, and this is denoted by these plus or minuses here.

01:06 So what we want to do is substitute both of these terms and our original expression into this equation with the substitution a equals pi over 4 as well as b equals s where we discern which version of the equation to use depending on whether or not these two arguments are added or subtracted in each term so let's start with the first term for the sign of pi over 4 plus s we want to use the plus version of this equation and what we get is the sign of a, which is pi over 4, times the cosine of b, which is s.

01:48 Added to that, we have the cosine of a, which is pi over 4, times the sign of b, which is s.

01:58 And now, subtracted from this, we have the second term, which can be expanded using the minus version of this equation.

02:06 We have the sign of a, which is pi over 4, times the cosine of b, which is s, minus.

02:15 The cosine of a, which is pi over 4, times the sign of b, which is s...

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