Find the exact value of each expression.

sin (13π/12)

Video transcript

Hey, everyone. Here we are asked to determine the exact value of the trigonometric expressions of 11 pi divided by 12. Here we have four answer choice options, answer choice A the square root of six plus the square root of two, all divided by four. Answer B the square root of six minus the square root of two, all divided by four. Answer C the square root of three divided by two and answer D the square root of six divided by two. So for this problem, we need to utilize the unit circle. And if we recall the unit circle, we know that we do not have this value 11 pi divided by 12. And so that tells us we need to split up this value into the sum of two terms which we can easily find on the unit circle. So that means that our current expression will become sine of pi divided by six plus three pi divided by four. Now that we have the sum of two terms, this tells us we need to recall the sum identity for the sine function which states that sine of A plus B is equivalent to sine of A multiplied by cosine of B plus cosine of A multiplied by sine of B. And now recalling our current expression so far, we have sine of pi divided by six plus three pi divided by four. And now comparing this expression to our formula, we can notice that our value for A is pi divided by six and our value for B is three pi divided by four. So now we can utilize the right hand side of the formula to expand our current expression. So, so far, we now have sine of pi divided by six, multiplied by cosine of three pi divided by four plus cosine of pi divided by six multiplied by sine of three pi divided by four. And now we just want to start solving the right hand side of our expression by first identifying each of the values for sine and cosine utilizing the unit circle. So we can start with our first term which is sine of pi divided by six. And now from the unit circle, we know that this is equivalent to one half. Next, we can look at cosine of pi divided by six. And we will see that this value is equivalent to the square root of three divided by two. Now we can move on to sine of three pi divided by four. And we will see that this value comes out to the square root of two divided by two. And lastly finding cosine of three pi divided by four, we see on the unit circle, we have negative square root of two divided by two. So now which with each of the values for Sine and cosine identified, we can go ahead and just substitute in these values into our expanded expression. And so we see that sign of divided by six plus three, pi divided by four is equivalent to one half multiplied by negative square root of two, divided by two plus the square root of three divided by two, multiplied by the square root of two divided by two. And now all we need to do is simplify by first multiplying the terms together. And so we have negative square root of two divided by four plus the square root of six divided by four. And as we can clearly see both of our fractions have common denominators, so we can finish simplifying by simply summing or combining the numerators into one fraction. And so we are left with the square root of six minus the square root of two, all divided by four. And now there are no further simplifications that we can make. So this is our final answer which leaves us with answer choice B where again, we have the square root of six minus the square root of two, all divided by four. Thanks for watching. I hope you found this video helpful and I'll see you again next time.