Find the exact value of each expression.
tan 285°

Question

Find the exact value of each expression.

tan 285°

Accepted Answer

Hey, everyone here we are asked to determine the exact value of the trigonometric expression tangent of 255 degrees. Here we have four answer choice options, answer choice A, the square root of two minus the square root of three. Answer B the square root of two plus three, answer C two minus the square root of three and answer D two plus the square root of three. So for this problem, we need to utilize our unit circle. And the first thing we can notice on our unit circle is that we do not have this 255 degree value. So that tells us we need to take this value and break it up into the sum of two values which we can find on the unit circle. So doing this, our expression becomes tangent of 120 degrees plus 135 degrees. And now that we have the sum of two terms in our tangent function, this tells us we need to recall the sum identity for tangent functions, which is tangent of A plus B is equivalent to tangent of A plus tangent of B all divided by one minus tangent of A multiplied by tangent of B. And now recalling our expanded expression so far, we have tangent of 120 degrees plus 135 degrees. And now comparing this expression to our formula, we can notice our value for A is 120 degrees and our value for B is 135 degrees. So now utilizing these values in the right hand side of the formula, we can expand our current expression further. And so we have tangent of 120 degrees plus tangent of 135 degrees, all divided by one minus tangent of 120 degrees multiplied by tangent of 135 degrees. So now with our fully expanded expression, we now want to solve for our tangent terms individually. So we can begin here with tangent of 120 degrees. And to solve for this term, we need to recall that tangent is equivalent to sine divided by cosine. So this tells us that tangent of 120 degrees is equivalent to sine of 120 degrees divided by cosine of 120 degrees. And now utilizing the unit circle, we can find these values directly for sine and cosine. And so we can rewrite our expression as the square root of three divided by two, divided by negative one half and now to make our current expression easier to simplify, we can rewrite it as multiplying by the reciprocal of the denominator. And so we have the square root of three divided by two, multiplied by negative two, divided by one. And now we can easily simplify here by noticing that the twos will cancel each other out. And so we are just left with negative square root of three. Now we can move on to our next term which is tangent of 135 degrees following the same steps as before, we can rewrite this term as sign of 135 degrees divided by cosine of 135 degrees. Once again, utilizing the unit circle, our expression now becomes the square root of two divided by two divided by negative square root of two divided by two. Now we can notice that the value in the numerator is the same as the denominator aside from the sign. So the values will cancel each other out and we are just left with negative one. And now that we have solved for both of our tangent terms, we can go back to our original expanded expression. So originally, again, we have tangent of 120 degrees plus 135 degrees is equivalent to this expanded expression form where we can now substitute in the values which we found. So we have negative square root of three plus negative one all divided by one minus negative square root of three multiplied by negative one. Now just simplifying we have negative square root of three minus one all divided by one minus the square root of three. Now, we aren't finished simplifying yet. Now, we need to rationalize our denominator. And so to rationalize the denominator, we simply multiply by its conjugate. So we multiply the numerator and the denominator by one plus the square root of three. And so now just multiplying out all of our terms, we have negative square root of three minus three minus one minus the square root of three, all divided by one plus the square root of three minus the square root of three minus three. And now just combining our like terms, we have negative four minus two multiplied by the square root of three divided by negative two. And now our last step in simplifying is first noticing that both of the terms in the numerator share a factor of negative two, which is the value in the denominator. So we can simply divide both of our terms by negative two. And we get our final simplified answer as two plus the square root of three. And with this final simplified answer, we are left here with answer choice D where again we have two plus the square root of three. 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.
tan 285°

Key Concepts

Reference Angles and Angle Reduction

Tangent Function and Its Sign in Quadrants

Using Angle Sum or Difference Identities

Watch next

Find the exact value of each expression.tan 285°

Key Concepts

Reference Angles and Angle Reduction

Tangent Function and Its Sign in Quadrants

Using Angle Sum or Difference Identities

Watch next

Find the exact value of each expression.
tan 285°