The expression you're asking about, \(\cos^{-1}\left(-\frac{1}{5}\right)\), represents the **inverse cosine** (or arccosine) of \(-\frac{1}{5}\).

The inverse cosine function, \(\cos^{-1}(x)\), gives the angle whose cosine is \(x\). We need to find the angle \(\theta\) such that:

\[
\cos(\theta) = -\frac{1}{5}
\]

The cosine of an angle is negative in the second and third quadrants of the unit circle. The range of the inverse cosine function is between \(0\) and \(\pi\) radians, so the angle \(\theta\) will be in the second quadrant.

Using a calculator:

\[
\cos^{-1}\left(-\frac{1}{5}\right) \approx 1.772 \text{ radians} \quad (\text{or approximately } 101.54^\circ)
\]

So, \(\cos^{-1}\left(-\frac{1}{5}\right) \approx 1.772\) radians.

Would you like further details or clarification?

Here are 5 related questions for further exploration:
1. What is the range and domain of the inverse cosine function?
2. How does the inverse cosine function differ from the regular cosine function?
3. How can you compute inverse trigonometric values without a calculator?
4. What are the properties of the cosine function in different quadrants?
5. How would you solve \(\cos^{-1}(x)\) for other values of \(x\)?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when working with trigonometric functions.

Discover how to calculate the inverse cosine of -1/5 using trigonometric principles. This solution includes an explanation of the range and properties of the cosine function, with a detailed breakdown of how to compute the answer in radians and degrees. Suitable for high school students in Grades 10-12.

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