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

Let's analyze whether \(\cos y > 0\), given the expressions for \(x\) and \(y\):

### Given:
- \(x = 2 \cos^{-1} \left(\frac{17}{35}\right)\)
- \(y = 2 \sin^{-1} \left(\frac{11}{21}\right)\)

We need to determine whether \(\cos y > 0\).

### Step 1: Analyze \(y = 2 \sin^{-1} \left(\frac{11}{21}\right)\)
- The principal value range for \(\sin^{-1}(z)\) is \(-\frac{\pi}{2} \leq \sin^{-1}(z) \leq \frac{\pi}{2}\).
- Thus, \(y = 2 \sin^{-1} \left(\frac{11}{21}\right)\) implies that \(y\) is in the range \(-\pi \leq y \leq \pi\).

### Step 2: Find the quadrant where \(y\) lies
Since \(\sin^{-1} \left(\frac{11}{21}\right)\) is positive and in the first quadrant, multiplying by 2 still places \(y\) within the first two quadrants (but closer to \(\frac{\pi}{2}\)).

- If \(\sin^{-1} \left(\frac{11}{21}\right)\) is positive, then \(y\) falls in the range \((0, \pi)\).

### Step 3: Determine \(\cos y\)
- For angles \(y\) in the first quadrant (\(0 < y < \frac{\pi}{2}\)), \(\cos y > 0\).
- For angles \(y\) in the second quadrant (\(\frac{\pi}{2} < y < \pi\)), \(\cos y < 0\).

However, because \(y\) is \(2 \sin^{-1} \left(\frac{11}{21}\right)\) and lies in the first quadrant (as \(\sin^{-1} \left(\frac{11}{21}\right)\) is less than \(\frac{\pi}{2}\)), \(y\) should still be in the first quadrant, so \(\cos y > 0\).

### Conclusion:
\(\cos y > 0\) for the given value of \(y\).

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Would you like a more detailed explanation or have any other questions? Here are 5 related questions that might interest you:

1. How do inverse trigonometric functions work with principal values?
2. What are the principal value ranges for different inverse trigonometric functions?
3. How do you determine the sign of trigonometric functions in different quadrants?
4. What is the importance of the principal branch in inverse trigonometric functions?
5. How can we geometrically interpret inverse trigonometric functions?

**Tip:** Understanding the principal values of inverse trigonometric functions is key to determining the correct range and sign of trigonometric expressions.

Explore the analysis of trigonometric functions in determining whether \(\cos y > 0\), given specific values for \(x\) and \(y\). This detailed solution covers the use of inverse trigonometric functions and their principal values, suitable for high school students.

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