Use the addition formulas for sine and cosine to simplify each expression. sin(t+(π)/(6))-sin(t-

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

Key Concepts

Trigonometric Addition Formulas

The trigonometric addition formulas allow the expression of the sine or cosine of a sum or difference of two angles in terms of the sines and cosines of the individual angles. These formulas, such as sin(a + b) = sin a cos b + cos a sin b and sin(a - b) = sin a cos b - cos a sin b, are foundational in simplifying trigonometric expressions and solving equations that involve composite angles.

Sum-to-Product Identities

Sum-to-product identities transform sums or differences of trigonometric functions into products. For example, the identity sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2) is used to simplify expressions where two sine functions with different arguments are subtracted. These identities are valuable for simplifying complex trigonometric expressions, solving equations, and analyzing wave behavior.

Special Angle Values

Special angle values refer to the trigonometric values at commonly used angles, such as ?/6, ?/4, ?/3, etc. Knowing these values, like sin(?/6) = 1/2, is crucial in simplifying expressions without resorting to a calculator. These standard values often emerge when applying the trigonometric addition and sum-to-product formulas, streamlining the simplification process in many problems.

Transcript

00:01 Hello everybody.

00:03 In this video, i'm going to be showing you how to solve exercise 22 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 quantity, sine of t plus pi over 6 minus sine.

00:23 Now, the first thing we want to do is recall the addition formulas for sign, which can be condensed in the following form.

00:30 For 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.

00:49 And so all we need to do is plug each of these two terms into this equation with the substitution, a equals t as well as b equals pi over six.

01:03 And then we also discern which version of the equation to use depending on whether or not the arguments in each sign term are added together or subtracted from each other.

01:13 Let's start with the first term, the sign of t plus pi over six.

01:17 Because these two angles are added together, we're going to use the plus version of this equation.

01:22 So the sign of t plus pi over six is equal to the sign of a, which is t, times the cosine of b, which is pi over six, plus the cosine of a, which is t, times the sign of b, which is pi over six.

01:42 And now subtracted from this, we have the second term.

01:44 Term, which we can expand using the minus version of this equation, as these two terms are subtracted from each other...

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