https://file.aiquickdraw.com/mathbot/file/1726289388906akw83wnh.jpg

In the image you've provided, we're tasked with several parts of trigonometric problems concerning a right triangle \( \triangle XYZ \) where \( \angle Z \) is a right angle.

### Part 1
Given that \( \angle X \) and \( \angle Y \) are complementary (since \( \angle Z \) is a right angle), \( \angle X + \angle Y = 90^\circ \). Thus, the following relationships hold:
- \( \sin X = \cos Y \)
- \( \cos X = \sin Y \)

### Part 2
Here, we should fill in the trigonometric functions based on the complementarity noted above:
- \( \sin X = \cos Y \)
- \( \cos X = \sin Y \)

### Part 3
We need to select true statements about the trigonometric functions:
- \( \cos X^2 = \cos Y \) is not generally true unless under specific angle conditions.
- \( \sin X = \sin Y \) and \( \cos X = \sin Y \) might be true depending on the specific angle but not universally so.
- \( \cos X = \sin Y \) is correct based on complementary angles.

### Part 4
To find \( \cos(66^\circ) = \sin(90^\circ - 66^\circ) = \sin(24^\circ) \). Thus, \( \cos(66^\circ) \) equals \( \sin(24^\circ) \).

If you need further clarification on any of these parts, or if you have more questions on this topic, please let me know! Here are some possible follow-up questions:
1. Can you explain why sine and cosine of complementary angles are equal?
2. What would \( \sin X \) and \( \cos Y \) be if \( X = 30^\circ \)?
3. How does the Pythagorean identity apply to this triangle?
4. Could you provide examples of using trigonometric identities in other contexts?
5. How do trigonometric functions apply to non-right triangles?

**Tip:** Remember, the cosine of any angle in a right triangle is equal to the sine of its complementary angle, which is helpful in simplifying expressions and solving problems.

In a right triangle XYZ with sides x, y, and z, we need to complete various trigonometric relationships involving angles X, Y, and Z, focusing on sine and cosine identities.

Learn how to solve trigonometric identities in a right triangle XYZ, where sin X = cos Y and cos X = sin Y. Explore the complementary relationship between sine and cosine for angles in a right triangle. Suitable for high school students in Grades 9-12.

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