To find the complement of [tex]\(\cot 30^\circ\)[/tex], let's follow these steps:
1. Understand the Relationship Between Tangent and Cotangent:
Cotangent ([tex]\(\cot\)[/tex]) is the reciprocal of the tangent ([tex]\(\tan\)[/tex]). In trigonometric terms, [tex]\(\cot(\theta) = \frac{1}{\tan(\theta)}\)[/tex].
2. Identify the Given Angle:
The given angle is [tex]\(30^\circ\)[/tex].
3. Find the Complementary Angle:
The complement of an angle [tex]\( \theta \)[/tex] is [tex]\(90^\circ - \theta\)[/tex]. Thus, the complement of [tex]\(30^\circ\)[/tex] is:
[tex]\[
90^\circ - 30^\circ = 60^\circ
\][/tex]
4. Determine the Tangent of the Complementary Angle:
We need to find [tex]\(\tan(60^\circ)\)[/tex]. According to trigonometric identities and values:
[tex]\(\tan(60^\circ)\)[/tex] is a well-known value:
[tex]\[
\tan(60^\circ) = \sqrt{3} \approx 1.7320508075688767
\][/tex]
Therefore, the complement of [tex]\(\cot 30^\circ\)[/tex] is [tex]\(\tan 60^\circ\)[/tex], and the steps lead us to the final answer:
- The complementary angle of [tex]\(30^\circ\)[/tex] is [tex]\(60^\circ\)[/tex].
- The value of [tex]\(\tan 60^\circ\)[/tex] is approximately [tex]\(1.7320508075688767\)[/tex].
So, the complement of [tex]\(\cot 30^\circ\)[/tex] is:
[tex]\[
60^\circ \quad \text{and} \quad 1.7320508075688767
\][/tex]
