Find the exact value of cot 300 degrees

Question

GPT-4o mini · Answer

To find the exact value of $\cot 300^\circ$, we can start by determining the reference angle and the quadrant in which $300^\circ$ lies. 1. **Finding the Reference Angle**: The reference angle for $300^\circ$ can be found as follows: $$ 300^\circ - 360^\circ = -60^\circ \quad \text{(so the reference angle is } 360^\circ - 300^\circ = 60^\circ\text{)} $$ 2. **Determining the Quadrant**: Since $300^\circ$ is in the fourth quadrant, the cotangent function is positive in that quadrant. 3. **Calculating $\cot 300^\circ$**: The cotangent function is defined as the reciprocal of the tangent function: $$ \cot \theta = \frac{1}{\tan \theta} $$ We focus on $\tan 60^\circ$: $$ \tan 60^\circ = \sqrt{3} $$ Hence, we find: $$ \cot 300^\circ = \cot(-60^\circ) = \frac{1}{\tan(-60^\circ)} = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} $$ Since $300^\circ$ is in the fourth quadrant, $\cot 300^\circ$ is positive: $$ \cot 300^\circ = \frac{1}{\sqrt{3}} $$ To present this in a simplified form, we multiply the numerator and denominator by $\sqrt{3}$: $$ \cot 300^\circ = \frac{\sqrt{3}}{3} $$ Thus, the exact value of $\cot 300^\circ$ is: $$ \boxed{\frac{\sqrt{3}}{3}} $$

oobleck · Answer

nope. In QIV, cot is negative. So cot 300° = -1/√3

GPT-4o mini · Answer

Apologies for the confusion in my previous response. You are correct; the cotangent function is indeed negative in the fourth quadrant.

To clarify the calculation of $\cot 300^\circ$:

The reference angle is $60^\circ$.
Since $300^\circ$ is in the fourth quadrant, where the cotangent function is negative, we have: \[ \tan 60^\circ = \sqrt{3} \]

Thus, we calculate: \[ \cot 300^\circ = \frac{1}{\tan 300^\circ} = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} \]

So, the correct value of $\cot 300^\circ$ is indeed: \[ \boxed{-\frac{1}{\sqrt{3}}} \]

Thank you for your patience, and I appreciate your correction!