Each expression is the right side of the formula for cos (α - ...

Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

Accepted Answer

Hello. Today we're gonna be finding the exact value of the following trigonometric expression. So we are given cosine of nine pi over 16, multiplied by cosine of five pi over plus sine of nine pi over 16, multiplied by sine of five pi over 16. Now, there is one way that we can simplify the following trigonometric sum and we can simplify it by using the difference property for cosine the difference property for cosine states that cosine of A minus B can be expanded as cosine of a multiplied by cosine of B plus sign of A multiplied by sign of B. So we can simplify the expression by using the difference property of cosine by using the property going from right to left. Now, we do need to identify what our AM B values are going to be. Now, one thing to note about the property is that the angle for the first cosine value and the first sign value are going to be the same and the angle for the second cosine value and second sine value are going to be the same. Notice that in the given expression, the angle for the first cosine value value is nine pi over 16. And the angle for the first sine value is nine pi over 16, that means that A is going to equal to nine pi over 16. And the remaining angle is going to be five pi over 16 for the second cosine value and five pi over 16 for the second sine value, that means B is going to equal to five pi over 16. So using the difference property for cosine, we can reduce the given expression as cosine nine pi over 16 minus five pi over 16, nine pi over 16 minus five, pi over 16 is going to reduce the cosine val value to cosine of four pi over 16 and four pi over 16 can be simplified to cosine of pi over four. What we'll need to do now is we'll need to evaluate the value of cosine pi over four. And in order to do that, we're going to use our unit circle. So we'll first need to figure out where pi over four exists on the unit circle. Now, if you're ever unsure of where radiant exists on the unit circle, you can always convert the radiant to degrees by multiplying the quantity by 180 over pi. This allows us to reduce the pies to just one. And what we are left with is 180 over four, which simplifies to 45. So that means that pie over four is equivalent to 45 degrees and 45 degrees on the unit circle is located in the first quadrant. And the terminal point of this of this angle is going to be square root 2/2 comma square root 2/2. Recall that any point on the unit circle is defined as cosine commas sign that means cosine represents any X value of any terminal point on the unit circle. And sine represents any Y value of any terminal point on the unit circle. Since we are looking for cosine pi over four, we want to pay attention to the given X value on our terminal point. And that is going to be square root 2/2. So that means cosine of pi over four is going to evaluate to square root 2/2. And this is going to be the exact value of the given trigonometric expression. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.
$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

Key Concepts

Cosine of a Difference Formula

Exact Values of Trigonometric Functions at Special Angles

Angle Simplification and Periodicity

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