https://file.aiquickdraw.com/mathbot/file/172630735694681fto847.jpg

Let's work through the questions shown in the image.

### Part 1: Fill in the blanks using \( T \), \( U \), and \( V \).
For the right triangle with side lengths \( T \), \( U \), and \( V \), where \( \angle L = 90^\circ \):

- \( \sin T = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{U}{V} \)
- \( \cos T = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{T}{V} \)
- \( \sin U = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{T}{V} \)
- \( \cos U = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{U}{V} \)

### Part 2: Identify the relationship between angles \( T \), \( U \), and \( L \).
Since this is a right triangle, \( \angle T \) and \( \angle U \) are complementary (their sum is \( 90^\circ \)).

Thus, in \( \triangle TUV \), \( \angle T \) and \( \angle U \) are **complementary**.

### Part 3: Select all true statements.
Let's verify each statement:
- \( \sin T = \sin U \): **False**. Since \( \sin T \) and \( \sin U \) are based on different angles, they cannot be equal.
- \( \sin T = \cos U \): **True**. Since \( \sin \theta = \cos (90^\circ - \theta) \), \( \sin T = \cos U \).
- \( \cos T = \cos U \): **False**.
- \( \cos T = \sin U \): **True**, as mentioned above for complementary angles.
- None of the above is true: **False**.

The true statements are \( \sin T = \cos U \) and \( \cos T = \sin U \).

### Part 4: Fill in the blank.
Given \( \sin(78^\circ) = \cos(\theta) \), use the complementary angle relationship \( \sin \theta = \cos (90^\circ - \theta) \).

Therefore, \( \theta = 12^\circ \), because \( 78^\circ + 12^\circ = 90^\circ \).

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Would you like further details on any part of the explanation, or do you have any other questions?

### Relative Questions:
1. What are the sine and cosine definitions for an angle in a right triangle?
2. How do complementary angles relate to sine and cosine?
3. Why are the sine and cosine of complementary angles equal to each other?
4. What are the steps to determine whether two angles in a right triangle are complementary?
5. How can we calculate the angle that complements a given angle in a right triangle?

### Tip:
Remember, sine and cosine of complementary angles are always equal, i.e., \( \sin \theta = \cos(90^\circ - \theta) \).

In a right triangle with side lengths T, U, and V, solve for the sine and cosine values of angles T and U, and determine the true trigonometric identities. Also, find the complementary angle for 78° using trigonometric relationships.

Solve trigonometric ratios for a right triangle, explore sine and cosine of angles T and U, and verify true trigonometric identities. Find complementary angles and apply basic trigonometric formulas. Suitable for high school students in Grades 9-10.

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