If Cot ⁡ Θ = − 3 \cot \theta = -\sqrt{3} Cot Θ = − 3 ​ And The Reference Angle Of Θ \theta Θ Is 30 ∘ 30^{\circ} 3 0 ∘ , Find Both Angles In Degrees For 0 ∘ ≤ Θ \textless 360 ∘ 0^{\circ} \leq \theta \ \textless \ 360^{\circ} 0 ∘ ≤ Θ \textless 36 0 ∘ And Both Angles In Radians For $0 \leq \theta

If $\cot \theta = -\sqrt{3}$ and the reference angle of $\theta$ is $30^{\circ}$, find both angles in degrees for $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$ and both angles in radians for $0 \leq \theta < 2\pi$

We are given that $\cot \theta = -\sqrt{3}$ and the reference angle of $\theta$ is $30^{\circ}$. Our goal is to find both angles in degrees for $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$ and both angles in radians for $0 \leq \theta < 2\pi$.

To solve this problem, we need to recall some trigonometric identities. The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle. We also know that the cotangent function is the reciprocal of the tangent function.

Since we are given that $\cot \theta = -\sqrt{3}$, we can use this information to find the value of $\theta$. We know that the cotangent function is negative in the second and fourth quadrants. Therefore, we can write:

$$\cot \theta = -\sqrt{3} = \cot (180^{\circ} + 30^{\circ}) = \cot (210^{\circ})$$

This means that $\theta$ is either $210^{\circ}$ or $330^{\circ}$.

To find the angles in radians, we can use the fact that $180^{\circ} = \pi$ radians. Therefore, we can convert the angles from degrees to radians as follows:

$$\theta = 210^{\circ} = \frac{7\pi}{6}$$

$$\theta = 330^{\circ} = \frac{11\pi}{6}$$

We have already found the angles in degrees as $210^{\circ}$ and $330^{\circ}$.

In this article, we have found both angles in degrees for $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$ and both angles in radians for $0 \leq \theta < 2\pi$ given that $\cot \theta = -\sqrt{3}$ and the reference angle of $\theta$ is $30^{\circ}$. We have used the cotangent function and trigonometric identities to solve this problem.

The final answer is $\boxed{210^{\circ}, 330^{\circ}, \frac{7\pi}{6}, \frac{11\pi}{6}}$.

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus

This article is for educational purposes only and is not intended to be used as a substitute for professional advice or guidance.
Q&A: If $\cot \theta = -\sqrt{3}$ and the reference angle of $\theta$ is $30^{\circ}$, find both angles in degrees for $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$ and both angles in radians for $0 \leq \theta < 2\pi$

A: The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle. It is the reciprocal of the tangent function.

A: We can use the fact that the cotangent function is negative in the second and fourth quadrants. Therefore, we can write:

$$\cot \theta = -\sqrt{3} = \cot (180^{\circ} + 30^{\circ}) = \cot (210^{\circ})$$

This means that $\theta$ is either $210^{\circ}$ or $330^{\circ}$.

A: We can use the fact that $180^{\circ} = \pi$ radians. Therefore, we can convert the angles from degrees to radians as follows:

$$\theta = 210^{\circ} = \frac{7\pi}{6}$$

$$\theta = 330^{\circ} = \frac{11\pi}{6}$$

A: The reference angles for the given angles are $30^{\circ}$.

A: We have already found the angles in degrees as $210^{\circ}$ and $330^{\circ}$.

A: We have already found the angles in radians as $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$.

A: The final answer is $\boxed{210^{\circ}, 330^{\circ}, \frac{7\pi}{6}, \frac{11\pi}{6}}$.

A: Some additional resources for learning more about trigonometry and calculus include:

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus
• [3] "Trigonometry" by Michael Corral
• [4] "Calculus" by Michael Spivak

A: No, this article is for educational purposes only and is not intended to be used as a substitute for professional advice or guidance.