Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. cos(45° - 30°)

Video transcript

Hello, today we're gonna be determining the exact value of the trigonometric expression. What we are given is cosine of 135 degrees minus 60 degrees. We can go ahead and expand the given expression by using the difference property of cosine the difference property of cosine states that if you have cosine A minus B, you can expand this expression as cosine of A multiplied by cosine of B plus sine of A multiplied by sign of B for a given expression, we're going to allow A to equal to 135 degrees and we're going to allow B to equal to 60 degrees. So utilizing the identity, we can expand our original expression as cosine of A which is 135 degrees multiplied by cosine of B which is degrees plus sine of A, which will be 135 degrees multiplied by sine of B which is 60 degrees. Now, what we'll need to do is we'll need to evaluate each of the trigonometric values. And in order to do that, we're going to need to use a unit circle. So let's first identify the locations of degrees and 60 degrees. 135 degrees is going to be located in the second quad quadrant. And the terminal point for 135 degrees is given to us as negative square root 2/2 comma positive square root 2/2 60 degrees will be located in the first quadrant of the unit circle. And the terminal point for this angle is going to be one half comma square root 3/2. Recall that any terminal point is expressed as cosine comma sign. What this means is that cosine is any X value for any terminal point and sine is any Y value for any terminal point. So in order to get the value of cosine 135 degrees, we're going to look at the X value for the terminal point of 135 degrees. That value is negative square root 2/2. So cosine of 135 degrees is going to be negative square root 2/2. Next for cosine of 60 degrees, we're going to take a look at the X value of the terminal point of 60 degrees. That value is given to us as positive one half. So cosine of 60 degrees will evaluate to give us one half. Next, we're gonna take a look at the Y value for the terminal point of 135 degrees and that is gonna be positive square root 2/2. So sine of 1 35 will evaluate to positive square root 2/2. And finally, sine of 60 degrees will be the Y value of the terminal point of 60 degrees which will be square root 3/2. So sinus 60 evaluates to square over 3/2. Now, what we need to do is we need to multiply the quantities together negative square root 2/2 0 multiplied by 1/2 will give us negative square root 2/4 and square root 2/2, multiplied by square root 3/2 will give us the value of square root 6/4. We can add these fractions together and combine them into a single fraction, which will be negative square root two plus square root of six all over four. And we can reorder the terms of the numerator going from positive to negative and we can rewrite the numerator as the square root of six minus the square root of two all over four. So what this means is that the exact value of cosine 1 minus 60 is going to equal to the square of six minus square of 2/4. And with that being said, the answer to this problem is going to be D. 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.