Let <A=90o , <B=47o , and side c have length of 7.7 cm.
Determine the length of side a to the nearest tenth of a unit

To find the length of side \( a \) in the given triangle with angles \( \angle A = 90^\circ \), \( \angle B = 47^\circ \), and side \( c \) with a length of 7.7 cm, we can use trigonometric relationships in a right triangle. Since \( \angle A \) is \( 90^\circ \), the given triangle is a right triangle, and we can use the following relationships:

1. **Sine** of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
   \]

2. **Cosine** of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
   \]

Here, the hypotenuse \( c \) is 7.7 cm. We need to find the length of side \( a \), which is opposite to angle \( B \). 

Using the sine function:
\[
\sin(B) = \frac{a}{c}
\]
Solving for \( a \):
\[
a = c \cdot \sin(B)
\]

Given:
\[
\sin(47^\circ) \approx 0.7314
\]

Therefore:
\[
a = 7.7 \cdot 0.7314 \approx 5.63
\]

To the nearest tenth of a unit, the length of side \( a \) is:
\[
a \approx 5.6 \text{ cm}
\]

Would you like more details on this calculation, or do you have any questions? Here are some related questions that might interest you:

1. How would you find the length of side \( b \) in this triangle?
2. What is the length of side \( b \) if \( \cos(47^\circ) \) is used?
3. Can you explain how the Pythagorean theorem applies to this problem?
4. What other trigonometric functions could be used to solve for side \( a \)?
5. How does the angle sum property of triangles verify the given angles?
6. What if angle \( B \) was different? How would that affect the length of side \( a \)?
7. How do you find the area of this triangle?
8. How would you solve this problem using a different method?

**Tip:** Remember, in a right triangle, knowing one acute angle and the hypotenuse allows you to find all the other sides and angles using basic trigonometric functions.

Learn how to calculate the length of side a in a right triangle with angles 90° and 47°, given side c = 7.7 cm. Use trigonometric ratios and the sine function to solve for the unknown side length. Suitable for high school students and those studying trigonometry.

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