Find the exact value.
(a)
cot((5/6) 𝜋)
(b)
cot((− 2/3) 𝜋)

To solve the cotangent of these angles, we can use the following identity for cotangent:

\[
\cot(\theta) = \frac{1}{\tan(\theta)}
\]

### (a) \(\cot\left(\frac{5}{6} \pi\right)\)
1. **Find the reference angle**:  
   \(\frac{5}{6} \pi = 150^\circ\).
2. **Tangent of 150°**:  
   Since \( \tan(150^\circ) = -\tan(30^\circ) = -\frac{1}{\sqrt{3}} \), the cotangent is:  
   \[
   \cot\left(\frac{5}{6} \pi\right) = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}
   \]
   So, \(\cot\left(\frac{5}{6} \pi\right) = -\sqrt{3}\).

### (b) \(\cot\left(-\frac{2}{3} \pi\right)\)
1. **Find the reference angle**:  
   \(-\frac{2}{3} \pi = -120^\circ\). In the third quadrant, where \(-120^\circ\) lies, tangent is positive.
2. **Tangent of 120°**:  
   \(\tan(120^\circ) = -\tan(60^\circ) = -\sqrt{3}\). Therefore, the cotangent is:  
   \[
   \cot\left(-\frac{2}{3} \pi\right) = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}}
   \]
   So, \(\cot\left(-\frac{2}{3} \pi\right) = -\frac{1}{\sqrt{3}}\).

### Final answers:
- (a) \(\cot\left(\frac{5}{6} \pi\right) = -\sqrt{3}\)
- (b) \(\cot\left(-\frac{2}{3} \pi\right) = -\frac{1}{\sqrt{3}}\)

Would you like further details on any step, or have any questions?

### 5 Related Questions:
1. What is the relationship between cotangent and sine/cosine?
2. How can we find the reference angle of a negative angle?
3. What is the value of \(\cot(\frac{7}{6}\pi)\)?
4. How can we determine the sign of trigonometric functions in different quadrants?
5. What is the exact value of \(\tan\left(\frac{11}{6}\pi\right)\)?

### Tip:
Always remember to determine in which quadrant the angle lies, as it affects the sign of trigonometric functions.

Find the exact value.
(a) cot((5/6) π)
(b) cot((− 2/3) π)

Learn how to find the exact values of cot(5/6 π) and cot(−2/3 π) step-by-step. This guide covers key trigonometric concepts like reference angles, cotangent, and quadrant-based function signs. Suitable for students in Grades 10-12.

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