Use the given information to find each of the following.

Question

cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°

Accepted Answer

Hello, today we're going to be determining the exact value of the given trigonometric expression using the half angle identities. So we are asked to determine the value of cotangent X divided by two. We are told that tangent of X is equal to the negative square root of 11 divided by six. And this is where the angle X lies between 90 degrees and 180 degrees. Now, this problem is particularly tricky because we're going to need to use different half angle identities and Pythagorean identities to help us determine the value of cotangent X divided by two. Let's quickly recall the Pythagorean identity for code tangent X divided by two. Now, cotangent of theta is equal to cosine of theta divided by sine of theta. Notice that in the Pythagorean identity, cotangent cosine and sine must all have the same angle theta. Well, in this case here, we want to identify the value of the half angle which is X divided by two. So we can rewrite this Pythia identity with the angle theta being X divided by two. And that will give us cotangent of X divided by two is equal to cosine of X divided by two divided by sine of X divided by two. What this means is that we need to determine the half angle values for cosine and sine in order to obtain the half angle value for cotangent. Now let's quickly recall the half angle identities for cosine and sine. The half angle identity for cosine X divided by two is equal to the positive or negative square root of one plus cosine theta divided by two. The half angle identity for sign X divided by two is equal to the positive or negative square root of one minus cosine theta divided by two. Now, in order to obtain the half angle values for cosine and sine, we must have a value for cosine. Let's quickly recall the Pythia identity for cosine. Now cosine theta is equal to one divided by seek theta. So how can we obtain this value of seek and data to obtain the value of cosine? Well, keep in mind that we are told that tangent of X is equal to the negative square root of 11 divided by six. There's a Pythagorean identity that includes both Sein and tangent. And that Pythagorean identity is seek and square theta is equal to one plus tangent squared data. So what this means is that we're going to start with this Python identity to obtain a value for sein, we'll plug in that value of seek to obtain a value for cosine. And once we have the value for cosine, we can plug it in to the half angle identities for cosine and sine and use those values to finally get the half angle value of cotangent. So let's go ahead and start by determining the value of sequent. If we were to solve. For second, in this Python identity, we'll get seek and theta is equal to the square root of one plus tangent squared. Theta. Now keep in mind that a square root always includes a positive or negative value. Now the question here is, will our s value be positive or negative? Well, keep in mind that we are told that the angle must be between 90 degrees and 180 degrees. This represents quadrant two on the unit circle and see is negative in quadrant two of the unit circle. What this means is that we'll be taking the negative square root for our seek in value. So second will be equal to the negative square root of one plus tangent squared. The now keep in mind that tangent is given to us as the negative square root of 11 divided by six. If we replace the value, we can rewrite sin. Theta is equal to the negative square root of one plus the negative square root of 11 divided by six. That quantity squared negative 11 to negative square root 11 divided by six squared will give us the value of 11 divided by 36. So this will equal to the negative square root of one plus 11 divided by 36 and one plus 11 divided by 36 will give us the value of 47 divided by 36. So this will equal to the negative square root of 47 divided by 36. We can further simplify this by taking the square root of both the numerator and denominator individually. And that will give us the negative square root of 47 divided by six. So seeking the is equal to this value, the native skirt route of 47 divided by six, we can take this value of sin and plug it into cosine in order to obtain that value. So cosine theta is equal to one divided by the negative square root of 47 divided by six. In order to simplify this quantity, we'll have to take the reciprocal of the denominator and multiply it to both the numerator and denominator. This will give us negative six divided by the square root of 47. And we're going to take this value and multiply it to both numerator and denominator. This will simplify the cosine value to give us negative six divided by the square root of 47. Now we cannot leave a square root in the denominator. So we must rationalize the denominator by multiplying both the numerator and denominator by the square root of 47. This will help simplify the value of cosine to be negative six multiplied by the square root of 47 all divided by 47. So this is going to be the value of cosine that we are now going to plug into the half angle values of cosine and sine. So let's go ahead and start that process. So again, the half angle value of cosine is equal to the square root of one plus the cosine value we saw for earlier which is negative six multiply the square root of 47 divided by 47. And this whole value will be divided by two. If we were to simplify the numerator, the numerator will simplify to give us 47 minus six multiplied by the square root of 47 divided by 47 all divided by two. And a trick to simplifying this complex fraction is to multiply the denominator in the numerator with the overall denominator 47 multiplied by two will give us the value of 94. So that means that the half angle value for cosine is going to simplify to 47 minus six multiplied by the square root of 47 divided by 94. Now keep in mind that a square root always produces a positive and negative value. So the question is, do we take the positive version of this solution or the negative version of the solution will recall that earlier, the value of X lies between the angles of 180 degrees and 90 degrees. But in this case here, our new angle is X divided by two. So we need to rewrite the inequality to include the value of two of X divided by two. In order to do that, we can divide the entire quan inequality by two. And that is going to give us 90 divided by two which is 45 180 divided by two, which is 90. And our angle X divided by two will lie between these two angles. Now, the values of 45 degrees and 90 degrees all lie within the first quadrant of the unit circle. What this means is that we're going to be taking the positive value of cosine X divided by two. Since cosine is positive in the first quadrant of the unit circle. So the half angle value for cosine is the square root of 47 minus six, multiplied by the square root of 47 divided by 94. Now we're going to repeat this process for the half angle value of sign. Now again, sine is equal to the positive or negative square root of one minus the negative value of six multiplied by the square root of 47 divided by 47. And all of this will be divided by two. If we simplify the numerator, the numerator will simplify to give us 47 plus six multiplied by the square root of 47 divided by 47. This whole quantity divided by two and just like cosine, we're going to combine the two new denominators together by multiplying 47 and two, that is going to give us the value of 94. So the sign will have a denominator of 94. So again, do we take the positive or negative value of this square root? While sine is going to lie within the same region as cosine and sine is positive within the first quadrant of the unit circle. So we'll be taking the positive value of the square root. So the half angle value for sine of X divided by two is equal to the square root of 47 plus six multiplied by the square root of 47 divided by 94. Now that we have the values of cosine and sine, we're going to plug this in to the Pythagorean identity for cotangent. And that will give us the half angle value for cotangent. So co tangent of X divided by two is equal to the cosine half angle value which is the square root of 47 minus six, multiplied by the square root of 47 divided by 94. And all of this will be divided by sine of X divided by two, which came out to be the square root of 47 plus six multiplied by the square root of 47 divided by 94. In order to simplify this, we'll be taking the ra excuse me, we'll be taking the reciprocal the denominator and multiplying it to put both the numerator and denominator doing so, will simplify the value of co tangent X divided by two into the square root of 47 minus six, divided by the square root of 47 divided by 47 plus six, multiplied by the square root of 47. And this is going to be the final value for the half angle of cotangent. 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.

Use the given information to find each of the following.
cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°

Key Concepts

Trigonometric Functions

Quadrants of the Unit Circle

Half-Angle Identities

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