Question

Find the exact value of the inverse cosecant, secant, and cotangent of (√3)/3
, if possible.

csc^−1((√3)/3)=

0 −π/4
−π/6
π/2
π/6
π/3
π/4
The exact value cannot be determined.


sec^−1((√3)/3)=

5π/6
π/4
π/2
0 3π/4
π/3
π/6
The exact value cannot be determined.


cot^−1((√3)/3)=

π/3
π/6
π/4
π/2
3π/4
0 5π/6
The exact value cannot be determined.

Answer 1

To find the exact values for the inverse cosecant, secant, and cotangent of \(\frac{\sqrt{3}}{3}\), we'll evaluate each one separately.

1. **Finding \(\csc^{-1}\left(\frac{\sqrt{3}}{3}\right)\)**:

The cosecant function is the reciprocal of the sine function. Therefore, we need to find the angle whose cosecant is \(\frac{\sqrt{3}}{3}\):

\[
\csc(\theta) = \frac{\sqrt{3}}{3} \implies \sin(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3}
\]

This value is not possible, as the sine function only ranges from -1 to 1. Therefore, we conclude:

\[
\csc^{-1}\left(\frac{\sqrt{3}}{3}\right) \text{ does not have a defined value.}
\]

**Answer:** The exact value cannot be determined.

2. **Finding \(\sec^{-1}\left(\frac{\sqrt{3}}{3}\right)\)**:

The secant function is the reciprocal of the cosine function. We are looking for the angle whose secant is \(\frac{\sqrt{3}}{3}\):

\[
\sec(\theta) = \frac{\sqrt{3}}{3} \implies \cos(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3}
\]

This value is also not possible, as the cosine function also only ranges from -1 to 1. Therefore, we conclude:

\[
\sec^{-1}\left(\frac{\sqrt{3}}{3}\right) \text{ does not have a defined value.}
\]

**Answer:** The exact value cannot be determined.

3. **Finding \(\cot^{-1}\left(\frac{\sqrt{3}}{3}\right)\)**:

The cotangent function is the reciprocal of the tangent function. We are looking for the angle whose cotangent is \(\frac{\sqrt{3}}{3}\):

\[
\cot(\theta) = \frac{\sqrt{3}}{3} \implies \tan(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3}
\]

The angle whose tangent is \(\sqrt{3}\) is \(\frac{\pi}{3}\):

\[
\cot^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{3}
\]

**Answer:** \(\cot^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{3}\).

### Summary of the Results:
- \(\csc^{-1}\left(\frac{\sqrt{3}}{3}\right)\): **The exact value cannot be determined.**
- \(\sec^{-1}\left(\frac{\sqrt{3}}{3}\right)\): **The exact value cannot be determined.**
- \(\cot^{-1}\left(\frac{\sqrt{3}}{3}\right)\): \(\frac{\pi}{3}\)