Question

use reference angles to find the exact value of the following expression.
cot 120°

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. cot 120° = -2
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expre
b. the answer is undefined.

Explanation:

Step1: Determine the reference angle

120° is in the second quadrant. The reference angle \( \theta' \) for an angle \( \theta \) in the second quadrant is \( 180^\circ - \theta \). So, for \( \theta = 120^\circ \), the reference angle \( \theta' = 180^\circ - 120^\circ = 60^\circ \).

Step2: Recall the sign of cotangent in the second quadrant

Cotangent is \( \cot\alpha=\frac{\cos\alpha}{\sin\alpha} \). In the second quadrant, cosine is negative and sine is positive, so cotangent is negative.

Step3: Find the value of \( \cot 60^\circ \)

We know that \( \cot 60^\circ=\frac{\cos 60^\circ}{\sin 60^\circ}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \) (rationalizing the denominator). But since 120° is in the second quadrant, \( \cot 120^\circ=-\cot 60^\circ \). Wait, there is a mistake in the initial option A. Let's recalculate:

Wait, \( \cot\alpha=\frac{\cos\alpha}{\sin\alpha} \), \( \cos 120^\circ=\cos(180 - 60)^\circ=-\cos 60^\circ=-\frac{1}{2} \), \( \sin 120^\circ=\sin(180 - 60)^\circ=\sin 60^\circ=\frac{\sqrt{3}}{2} \). Then \( \cot 120^\circ=\frac{\cos 120^\circ}{\sin 120^\circ}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3} \). But the option A has -2, which is wrong. Wait, maybe the original problem was \( \cot 120^\circ \), but let's check again.

Wait, maybe I made a mistake. Wait, \( \cot\theta=\frac{1}{\tan\theta} \), \( \tan 120^\circ=\tan(180 - 60)^\circ=-\tan 60^\circ=-\sqrt{3} \), so \( \cot 120^\circ=\frac{1}{\tan 120^\circ}=\frac{1}{-\sqrt{3}}=-\frac{\sqrt{3}}{3}\approx - 0.577 \), which is not -2. So the option A is incorrect. But maybe the problem was \( \cot 150^\circ \)? No, the problem is \( \cot 120^\circ \). Wait, maybe the user's option A is wrong, but let's follow the correct calculation.

Wait, the correct value of \( \cot 120^\circ \) is \( -\frac{\sqrt{3}}{3} \), but the option A has -2, which is incorrect. But maybe there is a mistake in the problem or the options. However, if we assume that maybe the angle is 150°? No, 120° is given. Wait, maybe the original problem was \( \cot 120^\circ \), but the option A is wrong. But according to the given options, if we have to choose, but the correct value is \( -\frac{\sqrt{3}}{3} \), but since the option A is given as -2, which is incorrect. Wait, maybe I made a mistake in the reference angle. Wait, 120° is in the second quadrant, reference angle 60°, cotangent is cos/sin, cos 120 is -1/2, sin 120 is √3/2, so cot 120 is (-1/2)/(√3/2) = -1/√3 = -√3/3 ≈ -0.577, which is not -2. So the correct answer should be that option A is wrong, but maybe the problem has a typo. But according to the given options, if we have to proceed, but the correct value is \( -\frac{\sqrt{3}}{3} \), but since the option A is -2, which is incorrect. Wait, maybe the angle is 135°? No, 135° has cot 135° = -1. No. Wait, maybe the problem is \( \cot 240^\circ \)? 240° is in the third quadrant, reference angle 60°, cot 240° = cot 60° = √3/3? No, 240°: cos 240 = -1/2, sin 240 = -√3/2, so cot 240 = (-1/2)/(-√3/2) = 1/√3 = √3/3. No. Wait, maybe the angle is 180° - 30° = 150°, cot 150° = (cos 150)/(sin 150) = (-√3/2)/(1/2) = -√3 ≈ -1.732, still not -2. Wait, maybe the angle is 120° but the function is sec or csc? No, the problem is cot 120°.

Wait, maybe the user made a mistake in the options, but according to the correct calculation, \( \cot 120^\circ = -\frac{\sqrt{3}}{3} \), so the option A is incorrect. But since the option A is selected, maybe there is a mistake. However, let's re-express the correct steps:

Step1: Find the quadrant of 120°

120° is between 90° and 180°, so it is in the second quadrant.

Step2: Find the reference angle

Reference angle \( = 180^\circ - 120^\circ = 60^\circ \).

Step3: Determine the sign of cotangent

In the second quadrant, \( \cos\theta < 0 \), \( \sin\theta > 0 \), so \( \cot\theta=\frac{\cos\theta}{\sin\theta}<0 \).

Step4: Calculate \( \cot 60^\circ \)

\( \cot 60^\circ=\frac{\cos 60^\circ}{\sin 60^\circ}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \).

Step5: Find \( \cot 120^\circ \)

Since \( \cot 120^\circ = -\cot 60^\circ \) (because of the sign in the second quadrant), \( \cot 120^\circ = -\frac{\sqrt{3}}{3} \).

But the option A has -2, which is incorrect. So there is a mistake in the options. However, if we assume that maybe the angle is 120° and the function is something else, but according to the problem, it's cot 120°.

But since the user provided option A as -2, maybe there is a mistake in the problem. But the correct value is \( -\frac{\sqrt{3}}{3} \).

Answer:

The correct value of \( \cot 120^\circ \) is \( -\frac{\sqrt{3}}{3} \), so the option A with -2 is incorrect. However, if we follow the given options and there is a mistake, but the correct answer should be \( \boxed{-\dfrac{\sqrt{3}}{3}} \) (but the option A is wrong).