To find the value of [tex]\(\cot(-390^\circ)\)[/tex], follow these steps:
1. Normalize the Angle:
- Angles can be normalized to lie within the range [tex]\([0^\circ, 360^\circ)\)[/tex] by adding or subtracting multiples of 360°.
- Starting with [tex]\(-390^\circ\)[/tex]:
[tex]\[
-390^\circ \mod 360^\circ = -30^\circ
\][/tex]
- Since [tex]\(-30^\circ\)[/tex] is still a negative angle, we can add 360° to get a positive equivalent angle:
[tex]\[
-30^\circ + 360^\circ = 330^\circ
\][/tex]
So [tex]\(-390^\circ\)[/tex] is equivalent to [tex]\(330^\circ\)[/tex].
2. Convert to Radians:
- Trigonometric functions often require the angle to be in radians. Convert [tex]\(330^\circ\)[/tex] to radians:
[tex]\[
330^\circ \times \frac{\pi}{180} = \frac{11\pi}{6} \approx 12.04277 \text{ radians}
\][/tex]
3. Calculate the Cotangent:
- Cotangent is the reciprocal of tangent. Therefore, we need to evaluate [tex]\(\tan(330^\circ)\)[/tex] first.
- However, directly using the result:
[tex]\[
\cot(330^\circ) = \frac{1}{\tan(330^\circ)} = -1.7320508075688776
\][/tex]
4. Interpret the Results:
- The value [tex]\(-1.7320508075688776\)[/tex] is not among the multiple-choice options given.
Therefore, the value of [tex]\(\cot(-390^\circ)\)[/tex] is not among the choices shown.
