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Express the lengths a and b shown in the figure in terms of 𝜃.

To express the lengths \( a \) and \( b \) in terms of the angle \( \theta \) and the hypotenuse (which is given as 23), we can use the trigonometric definitions for sine and cosine.

Given:
- Hypotenuse \( c = 23 \)
- Angle \( \theta \)

### 1. Length of side \( a \):
\( a \) is the side opposite to the angle \( \theta \). We use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{23}
\]
So, solving for \( a \):
\[
a = 23 \cdot \sin(\theta)
\]

### 2. Length of side \( b \):
\( b \) is the side adjacent to the angle \( \theta \). We use the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{23}
\]
So, solving for \( b \):
\[
b = 23 \cdot \cos(\theta)
\]

### Final expressions:
\[
a = 23 \cdot \sin(\theta)
\]
\[
b = 23 \cdot \cos(\theta)
\]

Would you like more details or have any questions?

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**Related questions:**
1. How can you find \( \theta \) if both \( a \) and \( b \) are known?
2. What are the values of \( a \) and \( b \) if \( \theta = 30^\circ \)?
3. How do you derive the Pythagorean theorem from the given expressions for \( a \) and \( b \)?
4. What is the relationship between \( a \) and \( b \) if \( \theta = 45^\circ \)?
5. Can you express \( \theta \) in terms of \( a \) and \( b \)?
6. How do the lengths \( a \) and \( b \) change as \( \theta \) approaches \( 0^\circ \)?
7. How do the lengths \( a \) and \( b \) change as \( \theta \) approaches \( 90^\circ \)?
8. What is the area of the triangle in terms of \( \theta \) and the hypotenuse?

**Tip:** Remember that the sum of the squares of the sides in a right triangle equals the square of the hypotenuse: \( a^2 + b^2 = c^2 \).

Learn how to express the lengths a and b in a right triangle, given the angle theta and the hypotenuse. This problem involves trigonometric concepts such as the sine and cosine functions, and the application of the Pythagorean theorem. Suitable for students in Grades 9-12.

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