Calculate the Sin A and Sin B in each triangle

To calculate \(\sin A\) and \(\sin B\) for each triangle, we'll use the basic trigonometric identity:

\[
\sin(\text{Angle}) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}
\]

Let's analyze the two triangles based on the image details:

### Left Triangle:
- Angle \(A\) is opposite the side labeled \(10 \, \text{cm}\).
- The hypotenuse is labeled \(14 \, \text{cm}\).
  
### Right Triangle:
- Angle \(B\) is opposite the side labeled \(5 \, \text{cm}\).
- The hypotenuse is labeled \(13 \, \text{cm}\).

### Calculating \(\sin A\) for the Left Triangle:
\[
\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{10 \, \text{cm}}{14 \, \text{cm}} = \frac{10}{14} = \frac{5}{7} \approx 0.714
\]

### Calculating \(\sin B\) for the Right Triangle:
\[
\sin B = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5 \, \text{cm}}{13 \, \text{cm}} \approx 0.385
\]

These are the values of \(\sin A\) and \(\sin B\) for the respective triangles.

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions you might find useful:
1. How would you calculate \(\cos A\) and \(\cos B\) for these triangles?
2. What is the tangent of angles \(A\) and \(B\) in these triangles?
3. How do the values of \(\sin A\) and \(\sin B\) compare if the triangles are similar?
4. What would be the sine of the complementary angles \(C\) in both triangles?
5. How does the Pythagorean Theorem confirm the relationships used in these sine calculations?

**Tip:** Always remember that the sine of an angle in a right triangle is equal to the opposite side divided by the hypotenuse, which is essential for solving various trigonometric problems.

Learn how to calculate \( \sin A \) and \( \sin B \) in two triangles using trigonometric identities and right triangle properties. Detailed step-by-step solution provided with numerical examples and explanations.

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