How does the solution change if cot⁡(x)=20cot(*x*)=20 instead of −20−20?

What is the principal value for arctan⁡(−120)arctan(−201​) in degrees?

### 1. **How does the solution change if \( \cot(x) = 20 \) instead of \( -20 \)?**

If the equation is \( \cot(x) = 20 \), follow a similar process as before:

#### Step 1: Express in terms of tangent
\[
\cot(x) = 20 \implies \frac{1}{\tan(x)} = 20 \implies \tan(x) = \frac{1}{20}
\]

#### Step 2: Solve for \( x \)
The general solution for \( \tan(x) = \frac{1}{20} \) is:
\[
x = \arctan\left(\frac{1}{20}\right) + n\pi, \quad n \in \mathbb{Z}
\]

#### Step 3: Calculate the principal value
Using a calculator or trigonometric tables:
\[
\arctan\left(\frac{1}{20}\right) \approx 0.04996 \text{ radians}
\]

#### Step 4: Write the general solution
Thus, the general solution becomes:
\[
x \approx 0.04996 + n\pi, \quad n \in \mathbb{Z}
\]

**Comparison:** 
- The principal value changes from \( -0.04996 \) radians (when \( \cot(x) = -20 \)) to \( 0.04996 \) radians (when \( \cot(x) = 20 \)).
- The general structure of the solution remains the same, with the principal value being different due to the sign change.

### 2. **What is the principal value for \( \arctan\left(-\frac{1}{20}\right) \) in degrees?**

To convert the principal value from radians to degrees:

\[
\text{Principal value} = -0.04996 \text{ radians}
\]

Using the conversion formula:
\[
\text{Degrees} = \text{Radians} \times \frac{180}{\pi}
\]

Calculate:
\[
-0.04996 \times \frac{180}{\pi} \approx -2.863 \text{ degrees}
\]

So, the principal value for \( \arctan\left(-\frac{1}{20}\right) \) is approximately \( -2.863^\circ \).

Would you like to explore these further or have any additional questions?

Here are 5 related questions you might consider:

1. How would the solution for \( \tan(x) = 2 \) differ from \( \tan(x) = \frac{1}{20} \)?
2. What is the geometric interpretation of the arctangent function?
3. How does the sign of the tangent function affect the quadrant in which the solution lies?
4. What are the implications of periodicity in trigonometric functions when solving equations?
5. How do the values of \( \arctan\left(\frac{1}{20}\right) \) and \( \arctan\left(-\frac{1}{20}\right) \) compare in the unit circle?

**Tip:** Always remember to convert angles between radians and degrees when necessary, especially in contexts where the unit of measurement is not specified.

Explore how the solution changes when the cotangent function is positive versus negative. Discover the principal value for arctan(-1/20) in degrees with step-by-step calculations and comparisons. Suitable for high school students studying trigonometry and inverse trigonometric functions.

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