Find the exact value of each expression.
tan (-7π/12)

Question

Find the exact value of each expression.

tan (-7π/12)

Accepted Answer

Hey, everyone. Here we are asked to determine the exact value of the trigonometric expression tangent of negative 19 pi divided by 12. Here we have four answer choice options, answer choice A two plus the square root of three B three plus the square root of two C, the square root of three divided by two. And answer D the square root of six plus the square root of two all divided by four. For this problem, we need to utilize the unit circle. And we can notice that negative 19 pi divided by 12 does not appear anywhere on the unit circle. And because this value is negative, this tells us that we need to split up the value into the difference of two different values which are present on the unit circle. So doing this, we have tangent of pi divided by four minus 11 pi divided by six. And now with this expression, we notice that we have the difference of two terms within the tangent function. So this tells us that we need to recall the difference identity for tangent functions which tells us that tangent of A minus B is equivalent to tangent of A minus tangent of B all divided by one plus tangent of A multiplied by tangent of B. And now recalling that the expression we are now working with is tangent of pi divided by four minus 11 pi divided by six. And now comparing to the left hand side of our formula, we can notice that the value for A is pi divided by four and the value for B is then 11 pi divided by six. So now with these values identified, we can utilize the right hand side of the formula to fully expand our current expression. So doing this, we have tangent of pi divided by four minus tangent of 11 pi divided by six, all divided by one plus tangent of pi divided by four multiplied by tangent of 11 pi divided by six. And now that we have fully expanded our expression, we want to begin evaluating it. So we can start by evaluating the individual tangent expressions. So we can begin here with tangent of pi divided by four. And to evaluate this expression, we need to recall that tangent is equivalent to sine divided by cosine. Therefore, we know that tangent of pi divided by four is equivalent to sine of pi divided by four, divided by cosine of pi divided by four. And now to find the values of sine and cosine, we need to utilize the unit circle. And from the unit circle, we know that this expression is equivalent to the square root of two divided by two, divided by the square root of two divided by two. And with this, we can notice we have the same exact value in both the numerator and the denominator. So they cancel each other out and we are just left with positive one. So now we can move on to our other term tangent of 11 pi divided by six. And we need to solve this expression just like we did before. So we have sine of 11 pi divided by six divided by cosine of 11 pi divided by six. Now, once again, we need to use the unit circle to find our values. And we have negative one half divided by the square root of three divided by two. And now to make this expression easier to simplify instead of dividing by a fraction, we can multiply by its reciprocal. So we have negative one half multiplied by two divided by the square root of three. Now, we can easily notice that the twos will cancel each other out. And so we have negative one divided by the square root of three. Now all that is left to do in simplifying is to rationalize the denominator. So we just multiply the numerator and the denominator by the square root of three. Now multiplying we have negative square root of three divided by three. And with this, we have found all of the necessary components for our expanded expression. And So now we know that tangent of pi divided by four minus 11 pi divided by six is equivalent to one minus negative square root of three divided by three all divided by one plus one, multiplied by negative square root of three divided by three. And now just simplifying the signs and multiplying we have one plus the square root of three divided by three all divided by one minus the square root of three divided by three. Now to continue simplifying, we once again need to rationalize the denominator where now we need to multiply by the conjugate of the denominator. So we multiply the numerator and the denominator of our expression by one plus the square root of three divided by three. And again, we do this to both the numerator and the denominator. And now just multiplying out each of the values, our current expression becomes one plus the square root of three divided by three plus the square root of three divided by three plus three, divided by nine all divided by one plus the square root of three, divided by three minus the square root of three divided by three minus three divided by nine. Now just combining all of our like terms and simplifying the fractions. So far, we will have four divided by three plus two, multiplied by the square root of three, divided by three, all divided by two, divided by three. And now, instead of M dividing by this fraction we can multiply by its reciprocal. So just rewriting our expression here, we have four plus two, multiplied by the square root of three all divided by three. And now we multiply by three, divided by two. And now we can notice that the threes will cancel each other. And we can also notice that both of the terms in the numerator are divisible by two. So just dividing here, we come up with our final simplified answer, which is two plus the square root of three. And with this final simplified answer, we are left here with answer choice. A where again, our final answer is two plus the square root of three. Thanks for watching. I hope you found this video helpful. I'll see you again next time.

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Find the exact value of each expression.tan (-7π/12)

Watch next

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Find the exact value of each expression.
tan (-7π/12)