In Exercises 1–26, find the exact value of each expression._cot⁻¹...

Question

In Exercises 1–26, find the exact value of each expression._cot⁻¹ (−√3)

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this question that states for the following trigonometric expression evaluate and get the answer as an exact value and radiance where the trigonometric expression given is the inverse co tangent of negative one. Now the choices provided are a five pi divided by four B pi divided by four C seven pi divided by four and D three pi divided by four. Now for a question like this, let's set up an if and then situation. So let's say if the inverse cotangent of negative one equals X, let's say then regular cotangent of X is equal to negative one. Now what does this mean? Well, code tangents of X can be rewritten as one divided by tangent of X because cotangent is just the reciprocal of tangent. So therefore, we have that tangent of X is equal to one divided by negative one. And all we did there was plug in our negative one answer. So another rule that we know is that tangent of X is equal to sine of X divided by cosine of X. So now we have to see with the information that we have, which quadrant are we in. So we have the tenure of X is equal to one divided by negative one. Therefore, our sign will be positive and our cosine will be negative. So where does that occur? Well, cosine of X is negative and sign of X is positive in one quadrant. So now you have, you have to think about a quadrant in which both of these scenarios are true. And this occurs in quadrants, two cosine of X is to the left of or of the Y axis. Therefore, it is negative and sin of X is above the X axis, therefore, it is positive. So now we talk about one final rule or something that we should note and that is the range of the co tangent function. And this will be very important as you'll see in a second. So cool tangent which is the function in our problem is defined for a range of zero two pie. So our answer has to fall within that range. And from zero pi that range includes the first quadrant and the second quadrant therefore answers like five pi divided by four, which is letter choice A and seven pie divided by four with it, which is choice C cannot B answers because they fall out of that range because five pi divided by four and seven pi divided by four are both greater than pi. Therefore, they fall out of the range and therefore, they cannot be answers, they shouldn't even be considered. So with that information in mind, now we can go ahead and turn to the unit circle and use it to our advantage. So recall from the unit circle, let's say three pi divided by four A theta of three pi divided by four has a point at the unit circle. And this is a value that we should be able to recall. That point says that we have an X value of negative square root of two divided by two comma and Y value of positive squared of two divided by two. And that's our point. Now remember that in the unit circle sign of theta is equal to the Y values and cosine of theta is equal to the X values. So that, that is how they are associated. So therefore, if we have tangent of three pi divided by four, we said that tangent of X is equal to sin of X divided by cosine of X. So and the numerator will have sine which is associated with the Y values are our, are our theta which we said was positive squared of two divided by two. And that goes in the numerator. And now the denominator will have the X component which is negative square root of two divided by two. And this can be rewritten as one divided by negative one, which is the answer that we were looking for or negative one. So since we said that tension of X is equal to one divided by negative one. And we just said that 10 of three pi divided by four is equal to one divided by negative one. We have that therefore, the X in our problem will be equal to three pi divided by four. And just like that, we have solved our question that matches up with answer choice D so we can see how important it is to be able to recall different values from the unit circle as well as knowing different ranges of our functions. Thanks to that range, we were able to eliminate two of the answer choices. And we can come across a point for example, seven pi divided by four that could have the same answer as our three pi divided by four, but it falls out of the range. So we must eliminate it. So I hope that was helpful. And until next time.

In Exercises 1–26, find the exact value of each expression._cot⁻¹ (−√3)

Key Concepts

Inverse Trigonometric Functions

Cotangent Function

Quadrants and Angle Values

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