Find the exact value of each expression.
sin (-13π/12)

Question

Find the exact value of each expression.

sin (-13π/12)

Accepted Answer

Hey, everyone. Here we are asked to determine the exact value of the trigonometric expression sine of negative 17 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 B, the square root of six minus the square root of two, all divided by four C, the square root of six divided by four D, the square root of three divided by two. For this problem, we need to recall the unit circle and we can notice that negative 17 pi divided by 12 is nowhere on the unit circle. And so we need to instead break up this negative number into the difference of two values which are on the unit circle. So doing this, our original expression becomes sine of pi divided by three minus seven pi divided by four. And now we notice that we have the difference between two values within a sine function. So that tells us we need to recall the difference identity for sine functions which states that sine of A minus B is equivalent to sine of A multiplied by cosine of B minus cosine of A multiplied by sine of B. And now just rewriting the expression we're working with, we have a sign of pi divided by three minus seven pi divided by four. And now comparing this expression with the left hand side of the formula, we can notice that A takes the value of pi divided by three and B is seven pi divided by four. So now with these values identified, we can go ahead and use the right hand side of the formula to fully expand the expression we're working on. So we have sign of pi divided by three multiplied pi cosine of seven pi divided by four minus cosine of pi divided by three multiplied by sine of seven pi divided by four. And now that we have fully expanded our expression, we want to start evaluating it and we can do this by first evaluating each of these sine and cosine components individually. So here we can start with sine of pi divided by three. And to evaluate this expression, we need to recall the unit circle where we can find this value directly. And from the unit circle, we know that this is equivalent to the square root of three divided by two. Now moving on to cosine of pi divided by three, this expression is equivalent to one half. Now moving on to cosine of seven pi divided by four. Again, the unit circle tells us that this is equivalent to the square root of two divided by two. Lastly, moving on to sine of seven pi divided by four. This is equivalent to negative square root of two divided by two. Now, with each of the components evaluated, we can substitute in their values into the expanded expression. So we know so far that we have sine of pi divided by three minus seven pi divided by four is equivalent to the square root of three divided by two, multiplied by the square root of two divided by two minus one half, multiplied by negative square of two divided by two. And now we just need to begin simplifying by first multiplying our values together. So from the first product, we get the square root of six divided by four, then we have minus negative square root of two divided by four. Simplifying these signs, we have the square root of six divided by four plus the square root of two divided by four. We can notice that both of the fractions have common denominators. So we can simply combine their numerators. And so we find our final simplified expression as the square root of six plus the square root of two all divided by four. And with this final expression, we are left with answer choice. A where again, we have the square root of six plus 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.

Find the exact value of each expression.
sin (-13π/12)

Key Concepts

Unit Circle and Angle Measurement

Reference Angles and Angle Reduction

Sine Function Properties and Identities

Watch next

Find the exact value of each expression.sin (-13π/12)

Key Concepts

Unit Circle and Angle Measurement

Reference Angles and Angle Reduction

Sine Function Properties and Identities

Watch next

Find the exact value of each expression.
sin (-13π/12)