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

Question

Find the exact value of each expression.

sin (π/12)

Accepted Answer

Hello, today we're going to be determining the exact value of the trigonometric expression. The expression we will be evaluating is sine of 17 pi divided by 12. Now, the current angle is in the form of radiance. If you're ever unsure where the location of a radiant is on the unit circle, you can always take the angle in form of radiance and multiply it by the quantity 180 divided by pi doing so will allow you to rewrite the angle in terms of radiance two degrees. So multiplying the two values together will give us the degree 255 degrees. So we can rewrite the trigonometric expression to be sign of 255 degrees. Now, the tricky part about this problem is that 255 degrees is not a standard angle on the unit circle. What we can do is we can rewrite 255 degrees to be a sum of standard angles on the unit circle. We will go ahead and let 255 equal to 45 plus 210. Or in other words, we are letting the angle 255 degrees be the sum of the standard angles 45 degrees and 210 degrees. So using this substitution, we can rewrite the expression one more time to be sign of 45 degrees plus 210 degrees. Now, in order to evaluate this problem, we're going to need to use the sum formula for sign the sum formula for sine is given to us as sine of X plus Y is equal to sin of X multiplied by cosine of Y plus cosine of X multiplied by sine of Y. We need to figure out our X and Y values. In order to use this identity, 45 degrees is in the spot of X. So we will allow X to equal to 45 degrees. Furthermore, 210 degrees is in the spot of Y. So Y will equal to 210 degrees. Now that we have the values for X and Y, we can rewrite the expression to be sine of 45 degrees plus 210 degrees is equal to sine of X, which will be sine of 45 degrees multiplied by cosine of Y which will be cosine of 210 degrees plus cosine of X, which would be cosine of 45 degrees multiplied by sine of Y which should be sine of 210 degrees. Now, what we need to do is we need to evaluate each of the cosine and sine functions. In order to do this, we're going to need the help of a unit circle. So there are two main angles that we are working with. The first angle is 45 degrees which is located in the first quadrant of the unit circle. The corresponding terminal point to this angle is the square root of two divided by two comma square root of two divided by two. The second angle we are working with is 210 degrees. 210 degrees is located in the third quadrant of the unit circle. And the corresponding terminal point to this angle is given to us as negative square root three divided by two comma negative one half. So recall that any terminal point on the unit circle can be represented as cosine comma sine. What this means is that the corresponding cosine values will equal to the X values on the terminal points. And the corresponding Y sine values will equal to the Y values of the terminal points. So using this sine of 45 degrees will equal to the Y value of the terminal point located at the 45 degree mark. That value is given to us as the square root of two divided by two. So sign of 45 degrees will evaluate to be square root of two divided by two. Continuing this process cosine of 210 degrees will equal to the X value at the 210 degree mark. That value is given to us as a negative square root three divided by two. So cosine of 210 will evaluate to give his negative square root three divided by two cosine of 45 degrees is the X value at the 45 degree mark. And that value is given to us as the square root two divided by two. And finally sign of 210 degrees is the Y value at the 210 degree mark, which is given to us as negative one half. Now, what we need to do is we need to algebraically simplify square root of two divided by two multiplied by negative square root. Three divided by two will leave us with the value of the negative square root six divided by four square root of two divided by two multiplied by negative half will leave us with the negative square root of two divided by four. We can combine both the fractions together to leave us with the negative square root of six minus the square root of two divided by four. And if we were to factor a negative one out of the numerator, we can rewrite the final answer to be negative square root of six plus the square root of two divided by four. This will be the final value for the original trigonometric expression. 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.

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

Key Concepts

Unit Circle

Angle Sum and Difference Identities

Exact Values of Trigonometric Functions

Watch next

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

Key Concepts

Unit Circle

Angle Sum and Difference Identities

Exact Values of Trigonometric Functions

Watch next

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