If x = 2cos inverse (17/35) and y = 2sin inverse (11/21) [inverse trigonometric function only take principal values] then is cosy>0

To determine whether \(\cos y > 0\), we first need to evaluate the principal values of the inverse trigonometric functions and analyze the angle \(y = 2\sin^{-1} \left(\frac{11}{21}\right)\).

### Step 1: Evaluate the range of \(\sin^{-1}(z)\)
The range of the inverse sine function, \(\sin^{-1}(z)\), is \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This means that the angle \(\sin^{-1} \left(\frac{11}{21}\right)\) will be in the first or fourth quadrant, where the sine function is non-negative.

Let:
\[
\theta = \sin^{-1} \left( \frac{11}{21} \right)
\]
Thus, \( \sin \theta = \frac{11}{21} \) and \( \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

### Step 2: Evaluate \(y = 2\sin^{-1} \left(\frac{11}{21}\right)\)
Since \( \theta \) is in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\), the angle \( y = 2\theta \) will be in the range \([- \pi, \pi]\). Now, depending on whether \( \theta \) is positive or negative, \( y \) could fall within different quadrants.

- If \(\theta\) is positive (since \(\sin^{-1} \left(\frac{11}{21}\right) > 0\)), then \(y = 2\theta\) will be in the first or second quadrant.
- In the first and second quadrants, \(\cos y > 0\) in the first quadrant and \(\cos y < 0\) in the second quadrant.

### Step 3: Determine the quadrant of \(y\)
Given that \( \sin^{-1} \left(\frac{11}{21}\right) \) is a positive angle (because \( \frac{11}{21} > 0 \)), \( \theta \) is in the first quadrant. Therefore, \(y = 2\theta\) is also a positive angle, but since \(y\) is double the angle, it may push the angle into the second quadrant.

- In the second quadrant, \( \cos y \) would be negative.
  
### Conclusion
Based on the analysis, \( y = 2\sin^{-1} \left(\frac{11}{21}\right) \) is in the second quadrant, where \( \cos y < 0 \).

Thus, the statement is false: \( \cos y > 0 \) is **not true**.

### Follow-up Questions:
1. How do you compute the exact value of \( \sin^{-1} \left(\frac{11}{21}\right) \)?
2. Can you verify in which quadrant a doubled angle from the inverse trigonometric function would land?
3. What are the properties of the cosine function in various quadrants?
4. How does the principal value of the inverse trigonometric function affect the angle's quadrant?
5. Could you determine the exact value of \( y \) numerically?

### Tip:
When working with inverse trigonometric functions, always carefully consider the range and principal values to determine the correct quadrant for your angles.

Explore how to determine if cosine is positive in inverse trigonometric functions with detailed steps and analysis. This problem involves evaluating inverse sine and cosine functions and understanding angle quadrants in trigonometry.

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