In Exercises 14–19, use a sum or difference formula to find the e...

Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression.

cos(45° + 30°)

Accepted Answer

Hello, today we're gonna be determining the value of the given expression. So what we are given is cosine of 60 degrees plus 45 degrees. We can expand this expression by using the sum property for cosine the sum property for cosine states that if you have cosine A plus B, this can be expanded as cosine of a multiplied by cosine of B minus sine of A multiplied by S sign of B. So we can use this identity to expand the given expression. We're going to allow A to equal to 60 degrees and we're going to allow B to equal to 45 degrees. And if we use the identity, we can expand the given expression as cosine of 60 degrees multiplied by cosine of 45 degrees minus sine of 60 degrees multiplied by sine of 45 degrees. So what we'll need to do is we'll need to use the unit circle to evaluate each of the trigonometric values. So let's go ahead and draw a unit circle. Now, what we wanna do is we want to identify the location of 60 degrees and 45 degrees, both 45 degrees and 60 degrees will be located in the first quadrant of the unit circle. The terminal point for 45 degrees is square root 2/ comma square root 2/2. And the terminal point for 60 degrees is going to be one half comma square root 3/2. Now recall that any terminal point on the unit circle is defined by cosine commas sign. This means that cosine represents any X value of any terminal point and sine represents any Y value of any terminal point. So with that being said, what is going to be the X value for cosine for 60 degrees? Well, cosine of 60 degrees is going to be one half. So we can replace this value with the half. Now, we want to evaluate cosine of 45 degrees. We're going to take a look at the terminal point of 45 degrees and the X value is square root 2/2. So we can replace cosine of 45 with the square root 2/2. Next, we're going to look at sin of 60 degrees. The terminal point for 60 degrees has a Y value of square root 3/2. So we could replace sine of 60 with square root 3/2. And finally, for sine of 45 degrees, we look at the Y value of the terminal point for 45 degrees and that is going to be square root 2/2. So a sin of 45 degrees will evaluate to give a square or 2/2. Now, what we wanna do is we want to multiply the values together and simplify one half, multiplied by square root 2/2, will give us the value of square root 2/4 and square root 3/2, multiplied by square root 2/2 will give us the value of the square root of 6/4. Finally, we can combine both of these fractions into a single fraction. And the final value is going to be square root two minus square root of six all over four. And what this means that the exact value of cosine plus 45 is going to equal to square root two minus square root of 6/4. And with that being said, the answer to this problem is going to be B 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.

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In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. cos(45° + 30°)