Find the exact value of each expression.
sin (- 5π/12)

Question

Find the exact value of each expression.

sin (- 5π/12)

Accepted Answer

Hey, everyone. Here we are asked to determine the exact value of the trigonometric expressions of negative 11 pi divided by 12. Here we have four answer choice options, answer choice A the square root of three plus the square root of two B, the square root of two divided by four C, the square root of six minus the square root of two all divided by four D negative square root of six plus the square root of two all divided by four. So to begin this problem, we need to recall the unit circle and notice that negative 11 pi divided by 12 does not appear on the unit circle. And so we need to split up this negative value into the difference of two values which appear on the unit circle. So our given expression becomes sine of pi divided by three minus five pi divided by four. And now we can notice we have the difference of two values within our sine function. So this tells us we need to recall the difference identity for a sine function 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 S of B. And now recalling that we are working with the expressions of pi divided by three minus five pi divided by four. And now comparing this expression to the left hand side of the formula, we can notice that our value for A is pi divided by three. And our value for B will then be five pi divided by four. So now using these values and the right hand side of the formula, we can go ahead and expand fully our expression so far. And so we have sine of pi divided by three, multiplied by cosine of five pi divided by four minus cosine of pi divided by three multiplied by sine of five pi divided by four. And now that we have fully expanded our original given expression, we can begin evaluating it by first evaluating each of the individual cosine and sine components. So here we can begin with sine of pi divided by three to find its value. We need to utilize the unit circle which states that this is equivalent to the square root of three divided by two. Now, if we look at the cosine for pi divided by three, we find that it is one half next going and looking at cosine of five pi divided by four, we know that its value is negative square root of two divided by two. And we also find that sine of five pi divided by four is the same. It is negative square root of two divided by two. Now, with each of the individual components evaluated, we can rewrite our expanded expression. So so far we know that sign of pi divided by three minus five, pi divided by four is equivalent to the square root of three divided by two, multiplied by the negative square root of two divided by two minus one half, multiplied by negative square root of two divided by two. And now we just want to multiply out each of our values here. So starting with our first product, we have the negative square root of six divided by four, we then have minus negative square root of two divided by four. Now just clean up the signs and combining the fractions, we find that our final solution is negative square root of six plus 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 D where again, we have negative 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 (- 5π/12)

Key Concepts

Unit Circle and Angle Measurement

Sine Function and Its Properties

Angle Sum and Difference Identities

Watch next

Find the exact value of each expression.sin (- 5π/12)

Key Concepts

Unit Circle and Angle Measurement

Sine Function and Its Properties

Angle Sum and Difference Identities

Watch next

Find the exact value of each expression.
sin (- 5π/12)